590
Chapter 5 | Integration
⎞ ⎠ |
⎛ ⎝ u 6 6
3
⎛ ⎝ 1 6
⎞ ⎠
3
1 6 ∫
u 5 du =
1
1
⎡ ⎣ (3) 6 −(1) 6 ⎤ ⎦
= 1
36
= 182 9 .
Use substitution to evaluate the definite integral ⌠ ⌡ −1 0 y ⎛
5.29
5
⎝ 2 y 2 −3 ⎞ ⎠
dy .
Example 5.35 Using Substitution with an Exponential Function
Use substitution to evaluate ∫ 0 1
2 +3
xe 4 x
dx .
Solution Let u =4 x 3 +3. Then, du =8 xdx . To adjust the limits of integration, we note that when x =0, u =3, and when x =1, u =7. So our substitution gives ∫ 0 1 xe 4 x 2 +3 dx = 1 8 ∫ 3 7 e u du = 1 8 e u | 3 7
= e 7 − e 3 8 ≈134.568.
⎛ ⎝ π 2
x 3 ⎞
Use substitution to evaluate ⌠ ⌡ 0 1
5.30
x 2 cos
⎠ dx .
Substitution may be only one of the techniques needed to evaluate a definite integral. All of the properties and rules of integration apply independently, and trigonometric functions may need to be rewritten using a trigonometric identity before we can apply substitution. Also, we have the option of replacing the original expression for u after we find the antiderivative, which means that we do not have to change the limits of integration. These two approaches are shown in Example 5.36 . Example 5.36
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