Indefinite Integrals

Differentiation allows us to gain a lot of information about a given function. Just as we can think of having a function f(x) and looking for its derivative, we can also consider going backwards - given a function f(x) we would like to find a function F (x) so that F 0(x) = f(x). Since this is the reverse process of differentiation, we call it antidifferentiation, and say that F is an antiderivative of f. Definition: Antiderivative A function F (x) is an antiderivative of f(x) if

F 0(x) = f(x)

for all x in the domain of f. We have seen many applications of differentiation, but a natural question to ask is why we are interested in antidifferentiation. One of the primary applications of antidifferentiation is in solving differential equations. Differential equations are equations that involve functions and their derivatives. Differential equations tend to govern most physical laws, so it is extremely worthwhile to be able to solve them. For instance, Newton’s second law states that an object’s mass times its acceleration is given by the sum of the forces acting on it (supposing that the mass of the object is constant), written F = ma If we think of acceleration as the second derivative of position, then we have the differential equation d2x F = m dx2 In classical mechanics we are interested in solving this differential equation for different physical situations. The solution to the above differential equation depends on what forces are at work in a given situation.

We have previously noted that when we differentiate a function, we lose any constant that is added to the function. Thus, if we have a single antiderivative to a given function, we can find another antiderivative by adding a constant to it. In fact, for F (x) an antiderivative of a function f(x), it turns out that every other antiderivative of f(x) can be written as F (x) + c for some constant c. Because we can choose any constant value for c it follows that this family of antiderivatives contains infinite members. This realization motivates the following definition. Definition: Indefinite Integral We call the set of all antiderivatives of f the indefinite integral of f, denoted by Z f(x)dx

The symbol R is an integral sign. The function f is the called the integrand and x is the variable of integration. We say that dx is a diﬀerential of x. Based on our previous discussion we can say that Z f(x)dx = F (x) + c because the expression on the right-hand side represents all possible antiderivatives of f(x), if we let c be an arbitrary constant.

When we find the indefinite integral of a function f(x) we say that we integrate the integrand f(x). Thus, the process of evaluating an integral is referred to as integration. Unfortunately, the notation we have introduced for the indefine integral probably looks rather arcane. As we move further into the study of integration, the reason for the notation will become more clear. The resemblence of R and an enlongated S is not accidental. This notation reflects the relationship between integration and summation. Integration is essentially the summation of the area underneath the integrand f(x) over intervals of infintessimal length. The differential dx represents an infintessimal change in x, which represents the intervals of infintessimal length over which the summation involved in integration occurs. The reason we have the notation dx is historical - calculus was initially conceived using infintessimals rather than limits. However, infintessimals at the time were not mathematically rigorous, but not we still study calculus using limits, because of the vast number of applications of limits (which we have yet begun to scratch the surface of). Finally, it is worth noting the resemblence in this dx and the derivative dy/dx. In fact, this notation is also historical, in considering an infintessimal change in y divided by an infintessimal change in x (this is analagous to considering the change in y over an interval as the length h in the limit as h → 0).

When it will not lead to confusion we will refer to the indefinite integral as simply the integral. The term indefinite integral is used to distinguish the process of indefinite and definite integration. Although both of these concepts are related to the area underneath a function, the indefinite integral is a function, whereas the definite integral is a constant, which is given by the area underneath a function over a set interval (defined by the limits of integration, which are not present in an indefinite integral).

Since the process of (indefinite) integration is an inverse to differentiation, we can derive many rules for integration using rules we already know for differentiation. For instance d xn+1 = (n + 1)xn dx so we see that d xn+1 = xn forx 6= −1 dx n + 1 This leads us to the product rule for integrals. Power Rule for Integrals

Z xn+1 xndx = + c n + 1 for every n 6= −1 (n ∈ R \ {−1}). Example 1 Evaluate the indeﬁnite integral of x2. Solution In this case we use the product rule, to see that

Z x2+1 x3 x2 = + c = + c 2 + 1 3 √ Example 2 Evaluate the indeﬁnite√ integral of x. Solution Once we rewrite x = x1/2 we see that

Z x1/2+1 2 x1/2 = + c = x3/2 + c 1/2 + 1 3

Example 3 Evaluate the indefinite integral of x−3. Solution We find that Z x−3+1 1 x−3 = + c = − x−2 + c −3 + 1 2 Example 4 Evaluate R dt. Solution Rewriting the integrand we find

Z Z t0+1 dt = t0dt = + c = t + c 0 + 1 Just like diﬀerentiation, integration is a linear operation. What this means is that integration satisﬁes the following two properties.

Sum Rule for Integrals Z Z Z f(x) + g(x)dx = f(x)dx + g(x)dx

Constant Product Rule for Integrals Z Z af(x)dx = a f(x)dx

for every a ∈ R

Example 5 Evaluate the indefinite integral of −2x−3 + 4x1/2 . Solution We can use the constant product rule and sum rules in conjunction to find Z Z Z 1 2 8 −2x−3 + 4x1/2 = −2 x−3 + 4 x1/2 = −2(− x−2) + 4( x3/2) + c = x−2 + x3/2 + c 2 3 3 In the above analysis we do not write the result as −2c + 4d, because we can easily enough choose c and d so that we have any value for the arbitrary constant. Thus, it is cleaner to just replace −2c + 4d with a single c, as both are capable of representing any arbitrary constant. Corresponding to other differentiation rules we have learned, we can evaluate the following integrals. Z exdx = ex + c Z cos(x)dx = sin(x) + c Z sin(x)dx = − cos(x) + c

Note that the negative sign corresponds to the integral of the sine function, just as the negative sign corresponds to the derivative of cosine. We’d also like to note Z 1 dx = ln(|x|) + c x

1 In the above formula we use the magnitude of x, |x|, because although x is deﬁned for negative x, ln(x) is not. We can verify this integral through diﬀerentiation. For x > 0

d d 1 ln(|x|) = ln(x) = dx dx x If we consider x < 0 then d d 1 1 ln(|x|) = ln(−x) = · (−1) = dx dx −x x We cannot evaluate the above derivative at x = 0, because ln(x) is undeﬁned for x = 0. Z 1 Example 6 Evaluate + 2ex + 3 cos(x) dx x Solution We can evaluate the above integral using the sum and constant product rules in conjunction with the integrals we have just discussed. Z 1 Z 1 Z Z + 2ex + 3 cos(x) dx = dx + 2 exdx + 3 cos(x)dx = ln(|x|) + 2ex + 3 sin(x) + c x x