Integration by Parts

The Idea Examples Nasty Example

Bernd Schroder¨

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 1. Simply put, symbolic integration is about undoing derivatives. 2. That is, we are given a function, which is actually a derivative f 0, and we are supposed to ﬁnd the original function f . 3. But most functions cannot be “obviously recognized” as derivatives of other functions. 4. “Reversing” differentiation rules provides methods to integrate symbolically. Note. Unlike differentiation, integration is not a cut-and-dried process. The integration methods may not give the solution right away, it won’t be clear which method to use (ﬁrst) and 2 there are even functions, like e−x , for which it can be proved that all symbolic integration methods fail.

The Idea Examples Nasty Example

Symbolic Integration

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 2. That is, we are given a function, which is actually a derivative f 0, and we are supposed to ﬁnd the original function f . 3. But most functions cannot be “obviously recognized” as derivatives of other functions. 4. “Reversing” differentiation rules provides methods to integrate symbolically. Note. Unlike differentiation, integration is not a cut-and-dried process. The integration methods may not give the solution right away, it won’t be clear which method to use (ﬁrst) and 2 there are even functions, like e−x , for which it can be proved that all symbolic integration methods fail.

The Idea Examples Nasty Example

Symbolic Integration 1. Simply put, symbolic integration is about undoing derivatives.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts , which is actually a derivative f 0, and we are supposed to ﬁnd the original function f . 3. But most functions cannot be “obviously recognized” as derivatives of other functions. 4. “Reversing” differentiation rules provides methods to integrate symbolically. Note. Unlike differentiation, integration is not a cut-and-dried process. The integration methods may not give the solution right away, it won’t be clear which method to use (ﬁrst) and 2 there are even functions, like e−x , for which it can be proved that all symbolic integration methods fail.

The Idea Examples Nasty Example

Symbolic Integration 1. Simply put, symbolic integration is about undoing derivatives. 2. That is, we are given a function

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts , and we are supposed to ﬁnd the original function f . 3. But most functions cannot be “obviously recognized” as derivatives of other functions. 4. “Reversing” differentiation rules provides methods to integrate symbolically. Note. Unlike differentiation, integration is not a cut-and-dried process. The integration methods may not give the solution right away, it won’t be clear which method to use (ﬁrst) and 2 there are even functions, like e−x , for which it can be proved that all symbolic integration methods fail.

The Idea Examples Nasty Example

Symbolic Integration 1. Simply put, symbolic integration is about undoing derivatives. 2. That is, we are given a function, which is actually a derivative f 0

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 3. But most functions cannot be “obviously recognized” as derivatives of other functions. 4. “Reversing” differentiation rules provides methods to integrate symbolically. Note. Unlike differentiation, integration is not a cut-and-dried process. The integration methods may not give the solution right away, it won’t be clear which method to use (ﬁrst) and 2 there are even functions, like e−x , for which it can be proved that all symbolic integration methods fail.

The Idea Examples Nasty Example

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 4. “Reversing” differentiation rules provides methods to integrate symbolically. Note. Unlike differentiation, integration is not a cut-and-dried process. The integration methods may not give the solution right away, it won’t be clear which method to use (ﬁrst) and 2 there are even functions, like e−x , for which it can be proved that all symbolic integration methods fail.

The Idea Examples Nasty Example

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Note. Unlike differentiation, integration is not a cut-and-dried process. The integration methods may not give the solution right away, it won’t be clear which method to use (ﬁrst) and 2 there are even functions, like e−x , for which it can be proved that all symbolic integration methods fail.

The Idea Examples Nasty Example

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Unlike differentiation, integration is not a cut-and-dried process. The integration methods may not give the solution right away, it won’t be clear which method to use (ﬁrst) and 2 there are even functions, like e−x , for which it can be proved that all symbolic integration methods fail.

The Idea Examples Nasty Example

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts The integration methods may not give the solution right away, it won’t be clear which method to use (ﬁrst) and 2 there are even functions, like e−x , for which it can be proved that all symbolic integration methods fail.

The Idea Examples Nasty Example

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts , it won’t be clear which method to use (ﬁrst) and 2 there are even functions, like e−x , for which it can be proved that all symbolic integration methods fail.

The Idea Examples Nasty Example

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts (ﬁrst) and 2 there are even functions, like e−x , for which it can be proved that all symbolic integration methods fail.

The Idea Examples Nasty Example

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts and 2 there are even functions, like e−x , for which it can be proved that all symbolic integration methods fail.

The Idea Examples Nasty Example

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts The Idea Examples Nasty Example

Symbolic Integration 1. Simply put, symbolic integration is about undoing derivatives. 2. That is, we are given a function, which is actually a derivative f 0, and we are supposed to find the original function f . 3. But most functions cannot be “obviously recognized” as derivatives of other functions. 4. “Reversing” differentiation rules provides methods to integrate symbolically. Note. Unlike differentiation, integration is not a cut-and-dried process. The integration methods may not give the solution right away, it won’t be clear which method to use (first) and 2 there are even functions, like e−x , for which it can be proved that all symbolic integration methods fail. logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 1. Integration by parts reverses the product rule. (fg)’(x)=f’(x)g(x)+g’(x)f(x) 2. Very few functions are sums of two products that fit the above pattern. 3. But products f (x)g(x) occur frequently.

The Idea Examples Nasty Example

Integration by Parts

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts (fg)’(x)=f’(x)g(x)+g’(x)f(x) 2. Very few functions are sums of two products that ﬁt the above pattern. 3. But products f (x)g(x) occur frequently.

The Idea Examples Nasty Example

Integration by Parts 1. Integration by parts reverses the product rule.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 2. Very few functions are sums of two products that ﬁt the above pattern. 3. But products f (x)g(x) occur frequently.

The Idea Examples Nasty Example

Integration by Parts 1. Integration by parts reverses the product rule. (fg)’(x)=f’(x)g(x)+g’(x)f(x)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 3. But products f (x)g(x) occur frequently.

The Idea Examples Nasty Example

Integration by Parts 1. Integration by parts reverses the product rule. (fg)’(x)=f’(x)g(x)+g’(x)f(x) 2. Very few functions are sums of two products that ﬁt the above pattern.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts The Idea Examples Nasty Example

Integration by Parts 1. Integration by parts reverses the product rule. (fg)’(x)=f’(x)g(x)+g’(x)f(x) 2. Very few functions are sums of two products that ﬁt the above pattern. 3. But products f (x)g(x) occur frequently.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Theorem. Integration by parts. Let f ,g be functions such that f has an antiderivative F and g is differentiable. If Z F(x)g0(x) dx exists, then Z Z f (x)g(x) dx = F(x)g(x) − F(x)g0(x) dx.

Proof. d Z F(x)g(x) − F(x)g0(x) dx dx = F0(x)g(x) + F(x)g0(x) − F(x)g0(x) = f (x)g(x).

The Idea Examples Nasty Example

Integration by Parts

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Integration by parts. Let f ,g be functions such that f has an antiderivative F and g is differentiable. If Z F(x)g0(x) dx exists, then Z Z f (x)g(x) dx = F(x)g(x) − F(x)g0(x) dx.

Proof. d Z F(x)g(x) − F(x)g0(x) dx dx = F0(x)g(x) + F(x)g0(x) − F(x)g0(x) = f (x)g(x).

The Idea Examples Nasty Example

Integration by Parts Theorem.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Let f ,g be functions such that f has an antiderivative F and g is differentiable. If Z F(x)g0(x) dx exists, then Z Z f (x)g(x) dx = F(x)g(x) − F(x)g0(x) dx.

Proof. d Z F(x)g(x) − F(x)g0(x) dx dx = F0(x)g(x) + F(x)g0(x) − F(x)g0(x) = f (x)g(x).

The Idea Examples Nasty Example

Integration by Parts Theorem. Integration by parts.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts If Z F(x)g0(x) dx exists, then Z Z f (x)g(x) dx = F(x)g(x) − F(x)g0(x) dx.

Proof. d Z F(x)g(x) − F(x)g0(x) dx dx = F0(x)g(x) + F(x)g0(x) − F(x)g0(x) = f (x)g(x).

The Idea Examples Nasty Example

Integration by Parts Theorem. Integration by parts. Let f ,g be functions such that f has an antiderivative F and g is differentiable.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts , then Z Z f (x)g(x) dx = F(x)g(x) − F(x)g0(x) dx.

Proof. d Z F(x)g(x) − F(x)g0(x) dx dx = F0(x)g(x) + F(x)g0(x) − F(x)g0(x) = f (x)g(x).

The Idea Examples Nasty Example

Integration by Parts Theorem. Integration by parts. Let f ,g be functions such that f has an antiderivative F and g is differentiable. If Z F(x)g0(x) dx exists

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Proof. d Z F(x)g(x) − F(x)g0(x) dx dx = F0(x)g(x) + F(x)g0(x) − F(x)g0(x) = f (x)g(x).

The Idea Examples Nasty Example

Integration by Parts Theorem. Integration by parts. Let f ,g be functions such that f has an antiderivative F and g is differentiable. If Z F(x)g0(x) dx exists, then Z Z f (x)g(x) dx = F(x)g(x) − F(x)g0(x) dx.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts d Z F(x)g(x) − F(x)g0(x) dx dx = F0(x)g(x) + F(x)g0(x) − F(x)g0(x) = f (x)g(x).

The Idea Examples Nasty Example

Proof.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts = F0(x)g(x) + F(x)g0(x) − F(x)g0(x) = f (x)g(x).

The Idea Examples Nasty Example

Proof. d Z F(x)g(x) − F(x)g0(x) dx dx

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts − F(x)g0(x) = f (x)g(x).

The Idea Examples Nasty Example

Proof. d Z F(x)g(x) − F(x)g0(x) dx dx = F0(x)g(x) + F(x)g0(x)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts = f (x)g(x).

The Idea Examples Nasty Example

Proof. d Z F(x)g(x) − F(x)g0(x) dx dx = F0(x)g(x) + F(x)g0(x) − F(x)g0(x)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts The Idea Examples Nasty Example

Proof. d Z F(x)g(x) − F(x)g0(x) dx dx = F0(x)g(x) + F(x)g0(x) − F(x)g0(x) = f (x)g(x).

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts The Idea Examples Nasty Example

Proof. d Z F(x)g(x) − F(x)g0(x) dx dx = F0(x)g(x) + F(x)g0(x) − F(x)g0(x) = f (x)g(x).

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 1. Integration by parts is a process, not a formula. 2. We choose a factor (call it f ) to be integrated and another factor (call it g) to be differentiated. Z Z f (x)g(x) dx = F(x)g(x) − F(x)g0(x) dx.

3. The choices must be guided by the fact that we must still integrate the product of the antiderivative of f and the derivative of g.

The Idea Examples Nasty Example

Integration by Parts

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts , not a formula. 2. We choose a factor (call it f ) to be integrated and another factor (call it g) to be differentiated. Z Z f (x)g(x) dx = F(x)g(x) − F(x)g0(x) dx.

3. The choices must be guided by the fact that we must still integrate the product of the antiderivative of f and the derivative of g.

The Idea Examples Nasty Example

Integration by Parts 1. Integration by parts is a process

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 2. We choose a factor (call it f ) to be integrated and another factor (call it g) to be differentiated. Z Z f (x)g(x) dx = F(x)g(x) − F(x)g0(x) dx.

3. The choices must be guided by the fact that we must still integrate the product of the antiderivative of f and the derivative of g.

The Idea Examples Nasty Example

Integration by Parts 1. Integration by parts is a process, not a formula.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Z Z f (x)g(x) dx = F(x)g(x) − F(x)g0(x) dx.

3. The choices must be guided by the fact that we must still integrate the product of the antiderivative of f and the derivative of g.

The Idea Examples Nasty Example

Integration by Parts 1. Integration by parts is a process, not a formula. 2. We choose a factor (call it f ) to be integrated and another factor (call it g) to be differentiated.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 3. The choices must be guided by the fact that we must still integrate the product of the antiderivative of f and the derivative of g.

The Idea Examples Nasty Example

Integration by Parts 1. Integration by parts is a process, not a formula. 2. We choose a factor (call it f ) to be integrated and another factor (call it g) to be differentiated. Z Z f (x)g(x) dx = F(x)g(x) − F(x)g0(x) dx.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts The Idea Examples Nasty Example

3. The choices must be guided by the fact that we must still integrate the product of the antiderivative of f and the derivative of g.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Integrate sin(x), differentiate x. Z Z xsin(x) dx = x − cos(x) − 1 · − cos(x) dx Z = −xcos(x) + cos(x) dx = −xcos(x) + sin(x) + c = sin(x) − xcos(x) + c Check. d √ ( sin(x)-x cos(x))=cos(x)- cos(x)-x(- sin(x))=x sin(x) dx

The Idea Examples Nasty Example Z Compute the Indeﬁnite Integral xsin(x) dx

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Z Z xsin(x) dx = x − cos(x) − 1 · − cos(x) dx Z = −xcos(x) + cos(x) dx = −xcos(x) + sin(x) + c = sin(x) − xcos(x) + c Check. d √ ( sin(x)-x cos(x))=cos(x)- cos(x)-x(- sin(x))=x sin(x) dx

The Idea Examples Nasty Example Z Compute the Indeﬁnite Integral xsin(x) dx Integrate sin(x), differentiate x.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Z = x − cos(x) − 1 · − cos(x) dx Z = −xcos(x) + cos(x) dx = −xcos(x) + sin(x) + c = sin(x) − xcos(x) + c Check. d √ ( sin(x)-x cos(x))=cos(x)- cos(x)-x(- sin(x))=x sin(x) dx

The Idea Examples Nasty Example Z Compute the Indeﬁnite Integral xsin(x) dx Integrate sin(x), differentiate x. Z xsin(x) dx

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Z − 1 · − cos(x) dx Z = −xcos(x) + cos(x) dx = −xcos(x) + sin(x) + c = sin(x) − xcos(x) + c Check. d √ ( sin(x)-x cos(x))=cos(x)- cos(x)-x(- sin(x))=x sin(x) dx

The Idea Examples Nasty Example Z Compute the Indeﬁnite Integral xsin(x) dx Integrate sin(x), differentiate x. Z xsin(x) dx = x − cos(x)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Z = −xcos(x) + cos(x) dx = −xcos(x) + sin(x) + c = sin(x) − xcos(x) + c Check. d √ ( sin(x)-x cos(x))=cos(x)- cos(x)-x(- sin(x))=x sin(x) dx

The Idea Examples Nasty Example Z Compute the Indeﬁnite Integral xsin(x) dx Integrate sin(x), differentiate x. Z Z xsin(x) dx = x − cos(x) − 1 · − cos(x) dx

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts = −xcos(x) + sin(x) + c = sin(x) − xcos(x) + c Check. d √ ( sin(x)-x cos(x))=cos(x)- cos(x)-x(- sin(x))=x sin(x) dx

The Idea Examples Nasty Example Z Compute the Indeﬁnite Integral xsin(x) dx Integrate sin(x), differentiate x. Z Z xsin(x) dx = x − cos(x) − 1 · − cos(x) dx Z = −xcos(x) + cos(x) dx

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts = sin(x) − xcos(x) + c Check. d √ ( sin(x)-x cos(x))=cos(x)- cos(x)-x(- sin(x))=x sin(x) dx

The Idea Examples Nasty Example Z Compute the Indeﬁnite Integral xsin(x) dx Integrate sin(x), differentiate x. Z Z xsin(x) dx = x − cos(x) − 1 · − cos(x) dx Z = −xcos(x) + cos(x) dx = −xcos(x) + sin(x) + c

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Check. d √ ( sin(x)-x cos(x))=cos(x)- cos(x)-x(- sin(x))=x sin(x) dx

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts d √ ( sin(x)-x cos(x))=cos(x)- cos(x)-x(- sin(x))=x sin(x) dx

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts √ =cos(x)- cos(x)-x(- sin(x))=x sin(x)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts √ =x sin(x)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts √

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts The Idea Examples Nasty Example Z Compute the Indeﬁnite Integral xsin(x) dx Integrate sin(x), differentiate x. Z Z xsin(x) dx = x − cos(x) − 1 · − cos(x) dx Z = −xcos(x) + cos(x) dx = −xcos(x) + sin(x) + c = sin(x) − xcos(x) + c Check. d √ ( sin(x)-x cos(x))=cos(x)- cos(x)-x(- sin(x))=x sin(x) dx

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Z 1 Z 1 x2e−4x dx = − e−4xx2 − − e−4x2x dx 4 4 1 1 Z = − e−4xx2 + e−4xx dx 4 2 1 1 1 Z 1 = − e−4xx2 + − e−4xx − − e−4x dx 4 2 4 4 1 1 1 Z = − e−4xx2 − e−4xx + e−4x dx 4 8 8 1 1 1 = − e−4xx2 − e−4xx − e−4x + c 4 8 32 1 1 1 = e−4x − x2 − x − + c 4 8 32

The Idea Examples Nasty Example Z Compute the Integral x2e−4x dx

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 1 Z 1 = − e−4xx2 − − e−4x2x dx 4 4 1 1 Z = − e−4xx2 + e−4xx dx 4 2 1 1 1 Z 1 = − e−4xx2 + − e−4xx − − e−4x dx 4 2 4 4 1 1 1 Z = − e−4xx2 − e−4xx + e−4x dx 4 8 8 1 1 1 = − e−4xx2 − e−4xx − e−4x + c 4 8 32 1 1 1 = e−4x − x2 − x − + c 4 8 32

The Idea Examples Nasty Example Z Compute the Integral x2e−4x dx

Z x2e−4x dx

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Z 1 − − e−4x2x dx 4 1 1 Z = − e−4xx2 + e−4xx dx 4 2 1 1 1 Z 1 = − e−4xx2 + − e−4xx − − e−4x dx 4 2 4 4 1 1 1 Z = − e−4xx2 − e−4xx + e−4x dx 4 8 8 1 1 1 = − e−4xx2 − e−4xx − e−4x + c 4 8 32 1 1 1 = e−4x − x2 − x − + c 4 8 32

The Idea Examples Nasty Example Z Compute the Integral x2e−4x dx

Z 1 x2e−4x dx = − e−4xx2 4

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 1 1 Z = − e−4xx2 + e−4xx dx 4 2 1 1 1 Z 1 = − e−4xx2 + − e−4xx − − e−4x dx 4 2 4 4 1 1 1 Z = − e−4xx2 − e−4xx + e−4x dx 4 8 8 1 1 1 = − e−4xx2 − e−4xx − e−4x + c 4 8 32 1 1 1 = e−4x − x2 − x − + c 4 8 32

The Idea Examples Nasty Example Z Compute the Integral x2e−4x dx

Z 1 Z 1 x2e−4x dx = − e−4xx2 − − e−4x2x dx 4 4

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 1 1 1 Z 1 = − e−4xx2 + − e−4xx − − e−4x dx 4 2 4 4 1 1 1 Z = − e−4xx2 − e−4xx + e−4x dx 4 8 8 1 1 1 = − e−4xx2 − e−4xx − e−4x + c 4 8 32 1 1 1 = e−4x − x2 − x − + c 4 8 32

The Idea Examples Nasty Example Z Compute the Integral x2e−4x dx

Z 1 Z 1 x2e−4x dx = − e−4xx2 − − e−4x2x dx 4 4 1 1 Z = − e−4xx2 + e−4xx dx 4 2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 1 Z 1 − e−4xx − − e−4x dx 4 4 1 1 1 Z = − e−4xx2 − e−4xx + e−4x dx 4 8 8 1 1 1 = − e−4xx2 − e−4xx − e−4x + c 4 8 32 1 1 1 = e−4x − x2 − x − + c 4 8 32

The Idea Examples Nasty Example Z Compute the Integral x2e−4x dx

Z 1 Z 1 x2e−4x dx = − e−4xx2 − − e−4x2x dx 4 4 1 1 Z = − e−4xx2 + e−4xx dx 4 2 1 1 = − e−4xx2 + 4 2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 1 1 1 Z = − e−4xx2 − e−4xx + e−4x dx 4 8 8 1 1 1 = − e−4xx2 − e−4xx − e−4x + c 4 8 32 1 1 1 = e−4x − x2 − x − + c 4 8 32

The Idea Examples Nasty Example Z Compute the Integral x2e−4x dx

Z 1 Z 1 x2e−4x dx = − e−4xx2 − − e−4x2x dx 4 4 1 1 Z = − e−4xx2 + e−4xx dx 4 2 1 1 1 Z 1 = − e−4xx2 + − e−4xx − − e−4x dx 4 2 4 4

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 1 1 1 = − e−4xx2 − e−4xx − e−4x + c 4 8 32 1 1 1 = e−4x − x2 − x − + c 4 8 32

The Idea Examples Nasty Example Z Compute the Integral x2e−4x dx

Z 1 Z 1 x2e−4x dx = − e−4xx2 − − e−4x2x dx 4 4 1 1 Z = − e−4xx2 + e−4xx dx 4 2 1 1 1 Z 1 = − e−4xx2 + − e−4xx − − e−4x dx 4 2 4 4 1 1 1 Z = − e−4xx2 − e−4xx + e−4x dx 4 8 8

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 1 1 1 = e−4x − x2 − x − + c 4 8 32

The Idea Examples Nasty Example Z Compute the Integral x2e−4x dx

Z 1 Z 1 x2e−4x dx = − e−4xx2 − − e−4x2x dx 4 4 1 1 Z = − e−4xx2 + e−4xx dx 4 2 1 1 1 Z 1 = − e−4xx2 + − e−4xx − − e−4x dx 4 2 4 4 1 1 1 Z = − e−4xx2 − e−4xx + e−4x dx 4 8 8 1 1 1 = − e−4xx2 − e−4xx − e−4x + c 4 8 32

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts The Idea Examples Nasty Example Z Compute the Integral x2e−4x dx

Z 1 Z 1 x2e−4x dx = − e−4xx2 − − e−4x2x dx 4 4 1 1 Z = − e−4xx2 + e−4xx dx 4 2 1 1 1 Z 1 = − e−4xx2 + − e−4xx − − e−4x dx 4 2 4 4 1 1 1 Z = − e−4xx2 − e−4xx + e−4x dx 4 8 8 1 1 1 = − e−4xx2 − e−4xx − e−4x + c 4 8 32 1 1 1 = e−4x − x2 − x − + c 4 8 32 logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Check.

d l l l e-4x - x2- x- +c dx ( 4 8 32) l l l l l = -4e-4x - x2- x- +e-4x - x- ( 4 8 32) ( 2 8) √ = e-4xx2

The Idea Examples Nasty Example Z Compute the Integral x2e−4x dx

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts d l l l e-4x - x2- x- +c dx ( 4 8 32) l l l l l = -4e-4x - x2- x- +e-4x - x- ( 4 8 32) ( 2 8) √ = e-4xx2

The Idea Examples Nasty Example Z Compute the Integral x2e−4x dx Check.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts l l l l l = -4e-4x - x2- x- +e-4x - x- ( 4 8 32) ( 2 8) √ = e-4xx2

The Idea Examples Nasty Example Z Compute the Integral x2e−4x dx Check.

d l l l e-4x - x2- x- +c dx ( 4 8 32)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts √ = e-4xx2

The Idea Examples Nasty Example Z Compute the Integral x2e−4x dx Check.

d l l l e-4x - x2- x- +c dx ( 4 8 32) l l l l l = -4e-4x - x2- x- +e-4x - x- ( 4 8 32) ( 2 8)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts √

The Idea Examples Nasty Example Z Compute the Integral x2e−4x dx Check.

d l l l e-4x - x2- x- +c dx ( 4 8 32) l l l l l = -4e-4x - x2- x- +e-4x - x- ( 4 8 32) ( 2 8) = e-4xx2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts The Idea Examples Nasty Example Z Compute the Integral x2e−4x dx Check.

d l l l e-4x - x2- x- +c dx ( 4 8 32) l l l l l = -4e-4x - x2- x- +e-4x - x- ( 4 8 32) ( 2 8) √ = e-4xx2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Z 1 Z 1 e2t sin(t) dt = e2t sin(t) − e2t cos(t) dt 2 2 1 1 1 Z 1 = e2t sin(t) − e2t cos(t) − e2t − sin(t) dt 2 2 2 2 1 1 1 Z = e2t sin(t) − e2t cos(t) − e2t sin(t) dt 2 4 4 5 Z 1 1 e2t sin(t) dt = e2t sin(t) − e2t cos(t) 4 2 4 Z 2 1 e2t sin(t) dt = e2t sin(t) − e2t cos(t) + c 5 5

The Idea Examples Nasty Example Z Compute the Integral e2t sin(t) dt

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 1 Z 1 = e2t sin(t) − e2t cos(t) dt 2 2 1 1 1 Z 1 = e2t sin(t) − e2t cos(t) − e2t − sin(t) dt 2 2 2 2 1 1 1 Z = e2t sin(t) − e2t cos(t) − e2t sin(t) dt 2 4 4 5 Z 1 1 e2t sin(t) dt = e2t sin(t) − e2t cos(t) 4 2 4 Z 2 1 e2t sin(t) dt = e2t sin(t) − e2t cos(t) + c 5 5

The Idea Examples Nasty Example Z Compute the Integral e2t sin(t) dt

Z e2t sin(t) dt

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Z 1 − e2t cos(t) dt 2 1 1 1 Z 1 = e2t sin(t) − e2t cos(t) − e2t − sin(t) dt 2 2 2 2 1 1 1 Z = e2t sin(t) − e2t cos(t) − e2t sin(t) dt 2 4 4 5 Z 1 1 e2t sin(t) dt = e2t sin(t) − e2t cos(t) 4 2 4 Z 2 1 e2t sin(t) dt = e2t sin(t) − e2t cos(t) + c 5 5

The Idea Examples Nasty Example Z Compute the Integral e2t sin(t) dt

Z 1 e2t sin(t) dt = e2t sin(t) 2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 1 1 1 Z 1 = e2t sin(t) − e2t cos(t) − e2t − sin(t) dt 2 2 2 2 1 1 1 Z = e2t sin(t) − e2t cos(t) − e2t sin(t) dt 2 4 4 5 Z 1 1 e2t sin(t) dt = e2t sin(t) − e2t cos(t) 4 2 4 Z 2 1 e2t sin(t) dt = e2t sin(t) − e2t cos(t) + c 5 5

The Idea Examples Nasty Example Z Compute the Integral e2t sin(t) dt

Z 1 Z 1 e2t sin(t) dt = e2t sin(t) − e2t cos(t) dt 2 2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 1 Z 1 e2t cos(t) − e2t − sin(t) dt 2 2 1 1 1 Z = e2t sin(t) − e2t cos(t) − e2t sin(t) dt 2 4 4 5 Z 1 1 e2t sin(t) dt = e2t sin(t) − e2t cos(t) 4 2 4 Z 2 1 e2t sin(t) dt = e2t sin(t) − e2t cos(t) + c 5 5

The Idea Examples Nasty Example Z Compute the Integral e2t sin(t) dt

Z 1 Z 1 e2t sin(t) dt = e2t sin(t) − e2t cos(t) dt 2 2 1 1 = e2t sin(t) − 2 2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 1 1 1 Z = e2t sin(t) − e2t cos(t) − e2t sin(t) dt 2 4 4 5 Z 1 1 e2t sin(t) dt = e2t sin(t) − e2t cos(t) 4 2 4 Z 2 1 e2t sin(t) dt = e2t sin(t) − e2t cos(t) + c 5 5

The Idea Examples Nasty Example Z Compute the Integral e2t sin(t) dt

Z 1 Z 1 e2t sin(t) dt = e2t sin(t) − e2t cos(t) dt 2 2 1 1 1 Z 1 = e2t sin(t) − e2t cos(t) − e2t − sin(t) dt 2 2 2 2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 5 Z 1 1 e2t sin(t) dt = e2t sin(t) − e2t cos(t) 4 2 4 Z 2 1 e2t sin(t) dt = e2t sin(t) − e2t cos(t) + c 5 5

The Idea Examples Nasty Example Z Compute the Integral e2t sin(t) dt

Z 1 Z 1 e2t sin(t) dt = e2t sin(t) − e2t cos(t) dt 2 2 1 1 1 Z 1 = e2t sin(t) − e2t cos(t) − e2t − sin(t) dt 2 2 2 2 1 1 1 Z = e2t sin(t) − e2t cos(t) − e2t sin(t) dt 2 4 4

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Z 2 1 e2t sin(t) dt = e2t sin(t) − e2t cos(t) + c 5 5

The Idea Examples Nasty Example Z Compute the Integral e2t sin(t) dt

Z 1 Z 1 e2t sin(t) dt = e2t sin(t) − e2t cos(t) dt 2 2 1 1 1 Z 1 = e2t sin(t) − e2t cos(t) − e2t − sin(t) dt 2 2 2 2 1 1 1 Z = e2t sin(t) − e2t cos(t) − e2t sin(t) dt 2 4 4 5 Z 1 1 e2t sin(t) dt = e2t sin(t) − e2t cos(t) 4 2 4

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts + c

The Idea Examples Nasty Example Z Compute the Integral e2t sin(t) dt

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts The Idea Examples Nasty Example Z Compute the Integral e2t sin(t) dt

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Check. d 2 l e2t sin(t)- e2t cos(t)+c dt(5 5 ) 4 2 2 l = e2t sin(t)+ e2t cos(t)- e2t cos(t)- e2t sin(t) 5 5 (5 5 ) √ = e2t sin(t)

The Idea Examples Nasty Example Z Compute the Integral e2t sin(t) dt

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts d 2 l e2t sin(t)- e2t cos(t)+c dt(5 5 ) 4 2 2 l = e2t sin(t)+ e2t cos(t)- e2t cos(t)- e2t sin(t) 5 5 (5 5 ) √ = e2t sin(t)

The Idea Examples Nasty Example Z Compute the Integral e2t sin(t) dt Check.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 4 2 2 l = e2t sin(t)+ e2t cos(t)- e2t cos(t)- e2t sin(t) 5 5 (5 5 ) √ = e2t sin(t)

The Idea Examples Nasty Example Z Compute the Integral e2t sin(t) dt Check. d 2 l e2t sin(t)- e2t cos(t)+c dt(5 5 )

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts √ = e2t sin(t)

The Idea Examples Nasty Example Z Compute the Integral e2t sin(t) dt Check. d 2 l e2t sin(t)- e2t cos(t)+c dt(5 5 ) 4 2 2 l = e2t sin(t)+ e2t cos(t)- e2t cos(t)- e2t sin(t) 5 5 (5 5 )

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts √

The Idea Examples Nasty Example Z Compute the Integral e2t sin(t) dt Check. d 2 l e2t sin(t)- e2t cos(t)+c dt(5 5 ) 4 2 2 l = e2t sin(t)+ e2t cos(t)- e2t cos(t)- e2t sin(t) 5 5 (5 5 ) = e2t sin(t)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts The Idea Examples Nasty Example Z Compute the Integral e2t sin(t) dt Check. d 2 l e2t sin(t)- e2t cos(t)+c dt(5 5 ) 4 2 2 l = e2t sin(t)+ e2t cos(t)- e2t cos(t)- e2t sin(t) 5 5 (5 5 ) √ = e2t sin(t)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Z Z arcsin(x) dx = 1 · arcsin(x) dx Z 1 = xarcsin(x) − x · √ dx 1 − x2 p = xarcsin(x) + 1 − x2 + c

The Idea Examples Nasty Example Z Compute the Integral arcsin(x) dx

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Z = 1 · arcsin(x) dx Z 1 = xarcsin(x) − x · √ dx 1 − x2 p = xarcsin(x) + 1 − x2 + c

The Idea Examples Nasty Example Z Compute the Integral arcsin(x) dx

Z arcsin(x) dx

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Z 1 = xarcsin(x) − x · √ dx 1 − x2 p = xarcsin(x) + 1 − x2 + c

The Idea Examples Nasty Example Z Compute the Integral arcsin(x) dx

Z Z arcsin(x) dx = 1 · arcsin(x) dx

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Z 1 − x · √ dx 1 − x2 p = xarcsin(x) + 1 − x2 + c

The Idea Examples Nasty Example Z Compute the Integral arcsin(x) dx

Z Z arcsin(x) dx = 1 · arcsin(x) dx

= xarcsin(x)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts p = xarcsin(x) + 1 − x2 + c

The Idea Examples Nasty Example Z Compute the Integral arcsin(x) dx

Z Z arcsin(x) dx = 1 · arcsin(x) dx Z 1 = xarcsin(x) − x · √ dx 1 − x2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts The Idea Examples Nasty Example Z Compute the Integral arcsin(x) dx

Z Z arcsin(x) dx = 1 · arcsin(x) dx Z 1 = xarcsin(x) − x · √ dx 1 − x2 p = xarcsin(x) + 1 − x2 + c

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Check. d √ (x arcsin(x)+ l-x2+c) dx l l l = arcsin(x)+x√ +( )√ (-2x) 2 2 2 √l-x l-x = arcsin(x)

The Idea Examples Nasty Example Z Compute the Integral arcsin(x) dx

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts d √ (x arcsin(x)+ l-x2+c) dx l l l = arcsin(x)+x√ +( )√ (-2x) 2 2 2 √l-x l-x = arcsin(x)

The Idea Examples Nasty Example Z Compute the Integral arcsin(x) dx Check.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts l l l = arcsin(x)+x√ +( )√ (-2x) 2 2 2 √l-x l-x = arcsin(x)

The Idea Examples Nasty Example Z Compute the Integral arcsin(x) dx Check. d √ (x arcsin(x)+ l-x2+c) dx

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts √ = arcsin(x)

The Idea Examples Nasty Example Z Compute the Integral arcsin(x) dx Check. d √ (x arcsin(x)+ l-x2+c) dx l l l = arcsin(x)+x√ +( )√ (-2x) l-x2 2 l-x2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts √

The Idea Examples Nasty Example Z Compute the Integral arcsin(x) dx Check. d √ (x arcsin(x)+ l-x2+c) dx l l l = arcsin(x)+x√ +( )√ (-2x) l-x2 2 l-x2 = arcsin(x)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts The Idea Examples Nasty Example Z Compute the Integral arcsin(x) dx Check. d √ (x arcsin(x)+ l-x2+c) dx l l l = arcsin(x)+x√ +( )√ (-2x) 2 2 2 √l-x l-x = arcsin(x)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Z p Z p p Z −x 1 − x2 dx = 1 · 1 − x2 dx = x 1 − x2 − x√ dx 1 − x2 p Z −x2 + 1 − 1 = x 1 − x2 − √ dx 1 − x2 p Z 1 Z 1 − x2 = x 1 − x2 + √ dx − √ dx 1 − x2 1 − x2 p Z p = x 1 − x2 + arcsin(x) − 1 − x2 dx Z p p 2 1 − x2 dx = x 1 − x2 + arcsin(x) Z p 1 p 1 1 − x2 = x 1 − x2 + arcsin(x) + c 2 2

The Idea Examples Nasty Example Z p Compute the Integral 1 − x2 dx

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Z p p Z −x = 1 · 1 − x2 dx = x 1 − x2 − x√ dx 1 − x2 p Z −x2 + 1 − 1 = x 1 − x2 − √ dx 1 − x2 p Z 1 Z 1 − x2 = x 1 − x2 + √ dx − √ dx 1 − x2 1 − x2 p Z p = x 1 − x2 + arcsin(x) − 1 − x2 dx Z p p 2 1 − x2 dx = x 1 − x2 + arcsin(x) Z p 1 p 1 1 − x2 = x 1 − x2 + arcsin(x) + c 2 2

The Idea Examples Nasty Example Z p Compute the Integral 1 − x2 dx

Z p 1 − x2 dx

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts p Z −x = x 1 − x2 − x√ dx 1 − x2 p Z −x2 + 1 − 1 = x 1 − x2 − √ dx 1 − x2 p Z 1 Z 1 − x2 = x 1 − x2 + √ dx − √ dx 1 − x2 1 − x2 p Z p = x 1 − x2 + arcsin(x) − 1 − x2 dx Z p p 2 1 − x2 dx = x 1 − x2 + arcsin(x) Z p 1 p 1 1 − x2 = x 1 − x2 + arcsin(x) + c 2 2

The Idea Examples Nasty Example Z p Compute the Integral 1 − x2 dx

Z p Z p 1 − x2 dx = 1 · 1 − x2 dx

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Z −x − x√ dx 1 − x2 p Z −x2 + 1 − 1 = x 1 − x2 − √ dx 1 − x2 p Z 1 Z 1 − x2 = x 1 − x2 + √ dx − √ dx 1 − x2 1 − x2 p Z p = x 1 − x2 + arcsin(x) − 1 − x2 dx Z p p 2 1 − x2 dx = x 1 − x2 + arcsin(x) Z p 1 p 1 1 − x2 = x 1 − x2 + arcsin(x) + c 2 2

The Idea Examples Nasty Example Z p Compute the Integral 1 − x2 dx

Z p Z p p 1 − x2 dx = 1 · 1 − x2 dx = x 1 − x2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts p Z −x2 + 1 − 1 = x 1 − x2 − √ dx 1 − x2 p Z 1 Z 1 − x2 = x 1 − x2 + √ dx − √ dx 1 − x2 1 − x2 p Z p = x 1 − x2 + arcsin(x) − 1 − x2 dx Z p p 2 1 − x2 dx = x 1 − x2 + arcsin(x) Z p 1 p 1 1 − x2 = x 1 − x2 + arcsin(x) + c 2 2

The Idea Examples Nasty Example Z p Compute the Integral 1 − x2 dx

Z p Z p p Z −x 1 − x2 dx = 1 · 1 − x2 dx = x 1 − x2 − x√ dx 1 − x2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts p Z 1 Z 1 − x2 = x 1 − x2 + √ dx − √ dx 1 − x2 1 − x2 p Z p = x 1 − x2 + arcsin(x) − 1 − x2 dx Z p p 2 1 − x2 dx = x 1 − x2 + arcsin(x) Z p 1 p 1 1 − x2 = x 1 − x2 + arcsin(x) + c 2 2

The Idea Examples Nasty Example Z p Compute the Integral 1 − x2 dx

Z p Z p p Z −x 1 − x2 dx = 1 · 1 − x2 dx = x 1 − x2 − x√ dx 1 − x2 p Z −x2 + 1 − 1 = x 1 − x2 − √ dx 1 − x2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts p Z p = x 1 − x2 + arcsin(x) − 1 − x2 dx Z p p 2 1 − x2 dx = x 1 − x2 + arcsin(x) Z p 1 p 1 1 − x2 = x 1 − x2 + arcsin(x) + c 2 2

The Idea Examples Nasty Example Z p Compute the Integral 1 − x2 dx

Z p Z p p Z −x 1 − x2 dx = 1 · 1 − x2 dx = x 1 − x2 − x√ dx 1 − x2 p Z −x2 + 1 − 1 = x 1 − x2 − √ dx 1 − x2 p Z 1 Z 1 − x2 = x 1 − x2 + √ dx − √ dx 1 − x2 1 − x2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Z p p 2 1 − x2 dx = x 1 − x2 + arcsin(x) Z p 1 p 1 1 − x2 = x 1 − x2 + arcsin(x) + c 2 2

The Idea Examples Nasty Example Z p Compute the Integral 1 − x2 dx

Z p Z p p Z −x 1 − x2 dx = 1 · 1 − x2 dx = x 1 − x2 − x√ dx 1 − x2 p Z −x2 + 1 − 1 = x 1 − x2 − √ dx 1 − x2 p Z 1 Z 1 − x2 = x 1 − x2 + √ dx − √ dx 1 − x2 1 − x2 p Z p = x 1 − x2 + arcsin(x) − 1 − x2 dx

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts p = x 1 − x2 + arcsin(x) Z p 1 p 1 1 − x2 = x 1 − x2 + arcsin(x) + c 2 2

The Idea Examples Nasty Example Z p Compute the Integral 1 − x2 dx

Z p Z p p Z −x 1 − x2 dx = 1 · 1 − x2 dx = x 1 − x2 − x√ dx 1 − x2 p Z −x2 + 1 − 1 = x 1 − x2 − √ dx 1 − x2 p Z 1 Z 1 − x2 = x 1 − x2 + √ dx − √ dx 1 − x2 1 − x2 p Z p = x 1 − x2 + arcsin(x) − 1 − x2 dx Z p 2 1 − x2 dx

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Z p 1 p 1 1 − x2 = x 1 − x2 + arcsin(x) + c 2 2

The Idea Examples Nasty Example Z p Compute the Integral 1 − x2 dx

Z p Z p p Z −x 1 − x2 dx = 1 · 1 − x2 dx = x 1 − x2 − x√ dx 1 − x2 p Z −x2 + 1 − 1 = x 1 − x2 − √ dx 1 − x2 p Z 1 Z 1 − x2 = x 1 − x2 + √ dx − √ dx 1 − x2 1 − x2 p Z p = x 1 − x2 + arcsin(x) − 1 − x2 dx Z p p 2 1 − x2 dx = x 1 − x2 + arcsin(x)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 1 p 1 = x 1 − x2 + arcsin(x) + c 2 2

The Idea Examples Nasty Example Z p Compute the Integral 1 − x2 dx

Z p Z p p Z −x 1 − x2 dx = 1 · 1 − x2 dx = x 1 − x2 − x√ dx 1 − x2 p Z −x2 + 1 − 1 = x 1 − x2 − √ dx 1 − x2 p Z 1 Z 1 − x2 = x 1 − x2 + √ dx − √ dx 1 − x2 1 − x2 p Z p = x 1 − x2 + arcsin(x) − 1 − x2 dx Z p p 2 1 − x2 dx = x 1 − x2 + arcsin(x) Z p 1 − x2 logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts The Idea Examples Nasty Example Z p Compute the Integral 1 − x2 dx

Z p Z p p Z −x 1 − x2 dx = 1 · 1 − x2 dx = x 1 − x2 − x√ dx 1 − x2 p Z −x2 + 1 − 1 = x 1 − x2 − √ dx 1 − x2 p Z 1 Z 1 − x2 = x 1 − x2 + √ dx − √ dx 1 − x2 1 − x2 p Z p = x 1 − x2 + arcsin(x) − 1 − x2 dx Z p p 2 1 − x2 dx = x 1 − x2 + arcsin(x) Z p 1 p 1 1 − x2 = x 1 − x2 + arcsin(x) + c 2 2 logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Check. d l √ l x l-x2+ arcsin(x)+c dx(2 2 ) l √ l x l l = l-x2- x√ + √ 2 2 l-x2 2 l-x2 l √ l l-x2 = l-x2+ √ 2 2 l-x2 √ √ = l-x2

The Idea Examples Nasty Example Z p Compute the Integral 1 − x2 dx

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts d l √ l x l-x2+ arcsin(x)+c dx(2 2 ) l √ l x l l = l-x2- x√ + √ 2 2 l-x2 2 l-x2 l √ l l-x2 = l-x2+ √ 2 2 l-x2 √ √ = l-x2

The Idea Examples Nasty Example Z p Compute the Integral 1 − x2 dx Check.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts l √ l x l l = l-x2- x√ + √ 2 2 l-x2 2 l-x2 l √ l l-x2 = l-x2+ √ 2 2 l-x2 √ √ = l-x2

The Idea Examples Nasty Example Z p Compute the Integral 1 − x2 dx Check. d l √ l x l-x2+ arcsin(x)+c dx(2 2 )

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts l √ l l-x2 = l-x2+ √ 2 2 l-x2 √ √ = l-x2

The Idea Examples Nasty Example Z p Compute the Integral 1 − x2 dx Check. d l √ l x l-x2+ arcsin(x)+c dx(2 2 ) l √ l x l l = l-x2- x√ + √ 2 2 l-x2 2 l-x2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts √ √ = l-x2

The Idea Examples Nasty Example Z p Compute the Integral 1 − x2 dx Check. d l √ l x l-x2+ arcsin(x)+c dx(2 2 ) l √ l x l l = l-x2- x√ + √ 2 2 l-x2 2 l-x2 l √ l l-x2 = l-x2+ √ 2 2 l-x2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts √

The Idea Examples Nasty Example Z p Compute the Integral 1 − x2 dx Check. d l √ l x l-x2+ arcsin(x)+c dx(2 2 ) l √ l x l l = l-x2- x√ + √ 2 2 l-x2 2 l-x2 l √ l l-x2 = l-x2+ √ 2 2 2 √ l-x = l-x2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts The Idea Examples Nasty Example Z p Compute the Integral 1 − x2 dx Check. d l √ l x l-x2+ arcsin(x)+c dx(2 2 ) l √ l x l l = l-x2- x√ + √ 2 2 l-x2 2 l-x2 l √ l l-x2 = l-x2+ √ 2 2 l-x2 √ √ = l-x2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Z te2t sin(t) dt 2 1 Z 2 1 = t e2t sin(t) − e2t cos(t) − e2t sin(t) − e2t cos(t) dt 5 5 5 5 2 1 2 Z 1 Z = te2t sin(t) − te2t cos(t) − e2t sin(t) dt + e2t cos(t) dt 5 5 5 5

The Idea Examples Nasty Example Z Compute the Integral te2t sin(t) dt

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 2 1 Z 2 1 = t e2t sin(t) − e2t cos(t) − e2t sin(t) − e2t cos(t) dt 5 5 5 5 2 1 2 Z 1 Z = te2t sin(t) − te2t cos(t) − e2t sin(t) dt + e2t cos(t) dt 5 5 5 5

The Idea Examples Nasty Example Z Compute the Integral te2t sin(t) dt

Z te2t sin(t) dt

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 2 1 2 Z 1 Z = te2t sin(t) − te2t cos(t) − e2t sin(t) dt + e2t cos(t) dt 5 5 5 5

The Idea Examples Nasty Example Z Compute the Integral te2t sin(t) dt

Z te2t sin(t) dt 2 1 Z 2 1 = t e2t sin(t) − e2t cos(t) − e2t sin(t) − e2t cos(t) dt 5 5 5 5

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts The Idea Examples Nasty Example Z Compute the Integral te2t sin(t) dt

Z te2t sin(t) dt 2 1 Z 2 1 = t e2t sin(t) − e2t cos(t) − e2t sin(t) − e2t cos(t) dt 5 5 5 5 2 1 2 Z 1 Z = te2t sin(t) − te2t cos(t) − e2t sin(t) dt + e2t cos(t) dt 5 5 5 5

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Z 1 Z 1 e2t cos(t) dt = e2t cos(t) + e2t sin(t) dt 2 2 1 1 1 Z 1 = e2t cos(t) + e2t sin(t) − e2t cos(t) dt 2 2 2 2 1 1 1 Z = e2t cos(t) + e2t sin(t) − e2t cos(t) dt 2 4 4 5 Z 1 1 e2t cos(t) dt = e2t cos(t) + e2t sin(t) 4 2 4 Z 2 1 e2t cos(t) dt = e2t cos(t) + e2t sin(t) 5 5

The Idea Examples Nasty Example Z Compute the Integral te2t sin(t) dt

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 1 Z 1 = e2t cos(t) + e2t sin(t) dt 2 2 1 1 1 Z 1 = e2t cos(t) + e2t sin(t) − e2t cos(t) dt 2 2 2 2 1 1 1 Z = e2t cos(t) + e2t sin(t) − e2t cos(t) dt 2 4 4 5 Z 1 1 e2t cos(t) dt = e2t cos(t) + e2t sin(t) 4 2 4 Z 2 1 e2t cos(t) dt = e2t cos(t) + e2t sin(t) 5 5

The Idea Examples Nasty Example Z Compute the Integral te2t sin(t) dt

Z e2t cos(t) dt

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 1 1 1 Z 1 = e2t cos(t) + e2t sin(t) − e2t cos(t) dt 2 2 2 2 1 1 1 Z = e2t cos(t) + e2t sin(t) − e2t cos(t) dt 2 4 4 5 Z 1 1 e2t cos(t) dt = e2t cos(t) + e2t sin(t) 4 2 4 Z 2 1 e2t cos(t) dt = e2t cos(t) + e2t sin(t) 5 5

The Idea Examples Nasty Example Z Compute the Integral te2t sin(t) dt

Z 1 Z 1 e2t cos(t) dt = e2t cos(t) + e2t sin(t) dt 2 2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 1 1 1 Z = e2t cos(t) + e2t sin(t) − e2t cos(t) dt 2 4 4 5 Z 1 1 e2t cos(t) dt = e2t cos(t) + e2t sin(t) 4 2 4 Z 2 1 e2t cos(t) dt = e2t cos(t) + e2t sin(t) 5 5

The Idea Examples Nasty Example Z Compute the Integral te2t sin(t) dt

Z 1 Z 1 e2t cos(t) dt = e2t cos(t) + e2t sin(t) dt 2 2 1 1 1 Z 1 = e2t cos(t) + e2t sin(t) − e2t cos(t) dt 2 2 2 2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 5 Z 1 1 e2t cos(t) dt = e2t cos(t) + e2t sin(t) 4 2 4 Z 2 1 e2t cos(t) dt = e2t cos(t) + e2t sin(t) 5 5

The Idea Examples Nasty Example Z Compute the Integral te2t sin(t) dt

Z 1 Z 1 e2t cos(t) dt = e2t cos(t) + e2t sin(t) dt 2 2 1 1 1 Z 1 = e2t cos(t) + e2t sin(t) − e2t cos(t) dt 2 2 2 2 1 1 1 Z = e2t cos(t) + e2t sin(t) − e2t cos(t) dt 2 4 4

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Z 2 1 e2t cos(t) dt = e2t cos(t) + e2t sin(t) 5 5

The Idea Examples Nasty Example Z Compute the Integral te2t sin(t) dt

Z 1 Z 1 e2t cos(t) dt = e2t cos(t) + e2t sin(t) dt 2 2 1 1 1 Z 1 = e2t cos(t) + e2t sin(t) − e2t cos(t) dt 2 2 2 2 1 1 1 Z = e2t cos(t) + e2t sin(t) − e2t cos(t) dt 2 4 4 5 Z 1 1 e2t cos(t) dt = e2t cos(t) + e2t sin(t) 4 2 4

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts The Idea Examples Nasty Example Z Compute the Integral te2t sin(t) dt

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Z te2t sin(t) dt 2 1 2 Z 1 Z = te2t sin(t) − te2t cos(t) − e2t sin(t) dt + e2t cos(t) dt 5 5 5 5 2 1 2 2 1 = te2t sin(t) − te2t cos(t) − e2t sin(t) − e2t cos(t) 5 5 5 5 5 1 2 1 + e2t cos(t) + e2t sin(t) 5 5 5 2 1 3 4 = te2t sin(t) − te2t cos(t) − e2t sin(t) + e2t cos(t) + c 5 5 25 25

The Idea Examples Nasty Example Z Compute the Integral te2t sin(t) dt

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 2 1 2 Z 1 Z = te2t sin(t) − te2t cos(t) − e2t sin(t) dt + e2t cos(t) dt 5 5 5 5 2 1 2 2 1 = te2t sin(t) − te2t cos(t) − e2t sin(t) − e2t cos(t) 5 5 5 5 5 1 2 1 + e2t cos(t) + e2t sin(t) 5 5 5 2 1 3 4 = te2t sin(t) − te2t cos(t) − e2t sin(t) + e2t cos(t) + c 5 5 25 25

The Idea Examples Nasty Example Z Compute the Integral te2t sin(t) dt

Z te2t sin(t) dt

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 2 1 2 2 1 = te2t sin(t) − te2t cos(t) − e2t sin(t) − e2t cos(t) 5 5 5 5 5 1 2 1 + e2t cos(t) + e2t sin(t) 5 5 5 2 1 3 4 = te2t sin(t) − te2t cos(t) − e2t sin(t) + e2t cos(t) + c 5 5 25 25

The Idea Examples Nasty Example Z Compute the Integral te2t sin(t) dt

Z te2t sin(t) dt 2 1 2 Z 1 Z = te2t sin(t) − te2t cos(t) − e2t sin(t) dt + e2t cos(t) dt 5 5 5 5

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 2 2 1 − e2t sin(t) − e2t cos(t) 5 5 5 1 2 1 + e2t cos(t) + e2t sin(t) 5 5 5 2 1 3 4 = te2t sin(t) − te2t cos(t) − e2t sin(t) + e2t cos(t) + c 5 5 25 25

The Idea Examples Nasty Example Z Compute the Integral te2t sin(t) dt

Z te2t sin(t) dt 2 1 2 Z 1 Z = te2t sin(t) − te2t cos(t) − e2t sin(t) dt + e2t cos(t) dt 5 5 5 5 2 1 = te2t sin(t) − te2t cos(t) 5 5

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 1 2 1 + e2t cos(t) + e2t sin(t) 5 5 5 2 1 3 4 = te2t sin(t) − te2t cos(t) − e2t sin(t) + e2t cos(t) + c 5 5 25 25

The Idea Examples Nasty Example Z Compute the Integral te2t sin(t) dt

Z te2t sin(t) dt 2 1 2 Z 1 Z = te2t sin(t) − te2t cos(t) − e2t sin(t) dt + e2t cos(t) dt 5 5 5 5 2 1 2 2 1 = te2t sin(t) − te2t cos(t) − e2t sin(t) − e2t cos(t) 5 5 5 5 5

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 2 1 3 4 = te2t sin(t) − te2t cos(t) − e2t sin(t) + e2t cos(t) + c 5 5 25 25

The Idea Examples Nasty Example Z Compute the Integral te2t sin(t) dt

Z te2t sin(t) dt 2 1 2 Z 1 Z = te2t sin(t) − te2t cos(t) − e2t sin(t) dt + e2t cos(t) dt 5 5 5 5 2 1 2 2 1 = te2t sin(t) − te2t cos(t) − e2t sin(t) − e2t cos(t) 5 5 5 5 5 1 2 1 + e2t cos(t) + e2t sin(t) 5 5 5

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts + c

The Idea Examples Nasty Example Z Compute the Integral te2t sin(t) dt

Z te2t sin(t) dt 2 1 2 Z 1 Z = te2t sin(t) − te2t cos(t) − e2t sin(t) dt + e2t cos(t) dt 5 5 5 5 2 1 2 2 1 = te2t sin(t) − te2t cos(t) − e2t sin(t) − e2t cos(t) 5 5 5 5 5 1 2 1 + e2t cos(t) + e2t sin(t) 5 5 5 2 1 3 4 = te2t sin(t) − te2t cos(t) − e2t sin(t) + e2t cos(t) 5 5 25 25

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts The Idea Examples Nasty Example Z Compute the Integral te2t sin(t) dt

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts Check. d 2 l 3 4 te2t sin(t)- te2t cos(t)- e2t sin(t)+ e2t cos(t) dx(5 5 25 25 ) 2 4 2 = e2t sin(t)+ te2t sin(t)+ te2t cos(t) 5 5 5 l 2 l - e2t cos(t)+ te2t cos(t)- te2t sin(t) (5 5 5 ) 6 3 - e2t sin(t)+ e2t cos(t) (25 25 ) 8 4 + e2t cos(t)- e2t sin(t) 25 25 √ = te2t sin(t)

The Idea Examples Nasty Example Z Compute the Integral te2t sin(t) dt

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts d 2 l 3 4 te2t sin(t)- te2t cos(t)- e2t sin(t)+ e2t cos(t) dx(5 5 25 25 ) 2 4 2 = e2t sin(t)+ te2t sin(t)+ te2t cos(t) 5 5 5 l 2 l - e2t cos(t)+ te2t cos(t)- te2t sin(t) (5 5 5 ) 6 3 - e2t sin(t)+ e2t cos(t) (25 25 ) 8 4 + e2t cos(t)- e2t sin(t) 25 25 √ = te2t sin(t)

The Idea Examples Nasty Example Z Compute the Integral te2t sin(t) dt Check.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 2 4 2 = e2t sin(t)+ te2t sin(t)+ te2t cos(t) 5 5 5 l 2 l - e2t cos(t)+ te2t cos(t)- te2t sin(t) (5 5 5 ) 6 3 - e2t sin(t)+ e2t cos(t) (25 25 ) 8 4 + e2t cos(t)- e2t sin(t) 25 25 √ = te2t sin(t)

The Idea Examples Nasty Example Z Compute the Integral te2t sin(t) dt Check. d 2 l 3 4 te2t sin(t)- te2t cos(t)- e2t sin(t)+ e2t cos(t) dx(5 5 25 25 )

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts l 2 l - e2t cos(t)+ te2t cos(t)- te2t sin(t) (5 5 5 ) 6 3 - e2t sin(t)+ e2t cos(t) (25 25 ) 8 4 + e2t cos(t)- e2t sin(t) 25 25 √ = te2t sin(t)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 6 3 - e2t sin(t)+ e2t cos(t) (25 25 ) 8 4 + e2t cos(t)- e2t sin(t) 25 25 √ = te2t sin(t)

The Idea Examples Nasty Example Z Compute the Integral te2t sin(t) dt Check. d 2 l 3 4 te2t sin(t)- te2t cos(t)- e2t sin(t)+ e2t cos(t) dx(5 5 25 25 ) 2 4 2 = e2t sin(t)+ te2t sin(t)+ te2t cos(t) 5 5 5 l 2 l - e2t cos(t)+ te2t cos(t)- te2t sin(t) (5 5 5 )

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts 8 4 + e2t cos(t)- e2t sin(t) 25 25 √ = te2t sin(t)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts √ = te2t sin(t)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts √

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts The Idea Examples Nasty Example Z Compute the Integral te2t sin(t) dt Check. d 2 l 3 4 te2t sin(t)- te2t cos(t)- e2t sin(t)+ e2t cos(t) dx(5 5 25 25 ) 2 4 2 = e2t sin(t)+ te2t sin(t)+ te2t cos(t) 5 5 5 l 2 l - e2t cos(t)+ te2t cos(t)- te2t sin(t) (5 5 5 ) 6 3 - e2t sin(t)+ e2t cos(t) (25 25 ) 8 4 + e2t cos(t)- e2t sin(t) 25 25 √ = te2t sin(t)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Integration by Parts