588
Chapter 5 | Integration
Use substitution to evaluate the integral ⌠ ⌡
5.27
cos t sin 2 t
dt .
Sometimes we need to manipulate an integral in ways that are more complicated than just multiplying or dividing by a constant. We need to eliminate all the expressions within the integrand that are in terms of the original variable. When we are done, u should be the only variable in the integrand. In some cases, this means solving for the original variable in terms of u . This technique should become clear in the next example. Example 5.33 Finding an Antiderivative Using u -Substitution Use substitution to find the antiderivative ∫ x x −1 dx . Solution If we let u = x −1, then du = dx . But this does not account for the x in the numerator of the integrand. We need to express x in terms of u . If u = x −1, then x = u +1. Now we can rewrite the integral in terms of u : ∫ x x −1 dx = ∫ u +1 u du = ∫ u + 1 u du = ∫ ⎛ ⎝ u 1/2 + u −1/2 ⎞ ⎠ du . Then we integrate in the usual way, replace u with the original expression, and factor and simplify the result. Thus, ∫ ⎛ ⎝ u 1/2 + u −1/2 ⎞ ⎠ du = 2 3 u 3/2 +2 u 1/2 + C = 2 3 ( x −1) 3/2 +2( x −1) 1/2 + C = ( x −1) 1/2 ⎡ ⎣ 2 3 ( x −1)+2 ⎤ ⎦ + C = ( x −1) 1/2 ⎛ ⎝ 2 3 x − 2 3 + 6 3 ⎞ ⎠
⎛ ⎝ 2 3
⎞ ⎠
= ( x −1) 1/2
x + 4 3
( x −1) 1/2 ( x +2)+ C .
= 2 3
Use substitution to evaluate the indefinite integral ∫ cos 3 t sin t dt .
5.28
Substitution for Definite Integrals Substitution can be used with definite integrals, too. However, using substitution to evaluate a definite integral requires a change to the limits of integration. If we change variables in the integrand, the limits of integration change as well.
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