ANTIDERIVATIVES

Definition:

reverse operation of finding a derivative Notice that F is called AN antiderivative and not THE antiderivative. This is easily understood by looking at the example above.

Some antiderivatives of 푓 푥 = 4푥3 are

퐹 푥 = 푥4, 퐹 푥 = 푥4 + 2, 퐹 푥 = 푥4 − 52

Because in each case

푑 퐹(푥) = 4푥3 푑푥 Theorem 1:

If a function has more than one antiderivative, then the antiderivatives differ by a constant.

• The graphs of antiderivatives are vertical translations of each other.

• For example: 푓(푥) = 2푥

Find several functions that are the antiderivatives for 푓(푥)

Answer: 푥2, 푥2 + 1, 푥2 + 3, 푥2 − 2, 푥2 + 푐 (푐 푖푠 푎푛푦 푟푒푎푙 푛푢푚푏푒푟) INDEFINITE INTEGRALS

Let f (x) be a function. The family of all functions that are antiderivatives of f (x) is called the indefinite integral and has the symbol  f (x) dx

The symbol  is called an integral sign, The function 푓 (푥) is called the integrand. The symbol 푑푥 indicates that anti-differentiation is performed with respect to the variable 푥. By the previous theorem, if 퐹(푥) is any antiderivative of 푓, then

f (x) dx  F(x) C  The arbitrary constant C is called the constant of integration. Indefinite Integral Formulas and Properties Vocabulary:

The indefinite integral of a function 푓(푥) is the family of all functions that are antiderivatives of 푓 (푥). It is a function 퐹(푥) whose derivative is 푓(푥).

The definite integral of 푓(푥) between two limits 푎 and 푏 is the area under the curve from 푥 = 푎 to 푥 = 푏. It is a number, not a function, equal to 퐹(푏) − 퐹(푎). Example 1:

푎. 2 푑푥 = 2푥 + 퐶

푏. 16 푒푡 푑푡 = 16 푒푡 + 퐶

푥5 3 c. 3푥4 푑푥 = 3 + 퐶 = 푥5 + 퐶 5 5 d.  (2x5  3x2 1)dx  2x5dx  3x2dx  1dx

6 3 5 2  x   x    2x dx  3x dx  1dx  2  3  1x  C  6   3  1  x6  x3  x  C 3  5  5 1 e.   4ex dx  dx  4exdx  5 dx  4 exdx  x   x   x   5ln x  4ex C

 2 3  2 2 f . 2x 3  dx  2x 3 dx  3x4dx  2 x 3 dx 3 x4dx  4       x 

 5   x 3   x3  6 5  2  3   C  x 3  x3  C  5   3  5      3  6 5 1  x 3   C 5 x3  x4 8x3  x3  x2  g.   dx  x2 8x dx  8   C  2        x  3  2  x3   4x2  C 3  4   1  1 1      6   x 3 x 2 h. 83 x  dx  8 x 3dx  6 x 2 dx  8   6   C  x    4   1         3   2  4  6x 3 12 x  C

i. (x2  2)(x 3)dx x3 3x2  2x 6dx 

x4   x3  x2  6x  C 4 Definition:

A differential equation is any equation which contains derivative(s). Solving a differential equation involves finding the original function from which the derivative came. The general solution involves C . The particular solution uses an initial condition to find the specific value of C.

Differential equation is called a separable differential equation if it is possible to separate 푥 and 푦 variables. If 푑푦 = 푓(푥) 푑푥 then the process of finding the antiderivatives of each side of the above equation (called indefinite integration) will lead to the solution.

푑푦 푑푥 = 푓(푥) 푑푥 ⇒ 푦 = 퐹 푥 + 퐶 푑푥 INITIAL VALUE PROBLEMS

Particular Solutions are obtained from initial conditions placed on the solution that will allow us to determine which solution that we are after.

Example: 푑푦 Solve the differential equation = 3푥2 if y 2 = −3. 푑푥 Find both the general and particular solution.

푑푦 푑푥 = 3푥2 푑푥 푑푥

general solution: 푦 = 푥3 + 퐶

particular solution: y 2 = −3 ⇒ −3 = 8 + 퐶 ⇒ 퐶 = −11

푦 = 푥3 − 11 Example:

Find the equation of the curve that passes through (2,6) if its slope is given by dy/dx = 3x2 at any point x.

The curve that has the derivative of 3x2 is  x3  2   3 3x dx 3   C  x  C  3 

Since we know that the curve passes through (2, 6), we can find out C

y  x3 C  6  23 C  C  2

Therefore, the equation is

푦 = 푥3 − 2