Section 4.5

The Substitution Rule

FINDING ANTIDERIVATIVES

Because of the Fundamental Theorem, it is important to be able to find antiderivatives. However, most antiderivatives are not easy to find. For example,

To do this, we need something extra. We change from the old variable to a new variable through the use of differentials.

dxxx 43 14

SUBSTITUTION

The process performed on the last problem is called u-substitution or simply substitution.

dxxgxgf )()(

Substitution will work anytime we have a composite function with the derivative of the “inside” function multiplied; that is, when we have the form

If F ′(x) = f (x), then

CxgFdxxgxgF )()()(

THE SUBSTITUTION RULE

Theorem: If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then

duufdxxgxgf )()()(

The Substitution Rule says: It is permissible to operate with dx and du after the integral as if they were differentials.

THE SUBSTITUTION RULE FOR DEFINITE INTEGRALS

Theorem: If g′ is continuous on [a, b] and f is continuous on the range of u = g(x), then

)(

)()()()(

bg

ag

b

aduufdxxgxgf

Theorem: Suppose f is continuous on [−a, a].

(a) If f is even (symmetric about the y-axis), then

(b) If f is odd (symmetric about the origin), then

INTEGRALS OF SYMMETRIC FUNCTIONS

.)(2)(0

aa

adxxfdxxf

.0)( a

adxxf