(6/12/08)
Math 10B. Lecture Examples.
Section 7.1. Integration by substitution†
Example 1 Find the antiderivative Z(x2+ 1)5(2x)dx.
Answer: Z(x2+ 1)5(2x)dx =1
6(x2+ 1)6+C
Example 2 Perform the integration Zx3√x4+ 16 dx.
Answer: Zx3√x4+ 16 dx =1
6(x4+ 16)3/2+C
Example 3 Find the antiderivatives Zcos(√x)
√xdx.
Answer: Zcos(√x)
√x
dx = 2 sin √x+C
Example 4 Figure 1 shows the region between y=10x
(x2+ 1)2and the x-axis for
0≤x≤3. Find its area.
x123
y
1
2
3
y=10x
(x2+ 1)2
FIGURE 1
Answer: [Area] = 9
2
Example 5 Find the value of Z1
0
sin(πx)dx.
Answer: Z1
0
sin(πx)dx =2
π
Example 6 Evaluate Z1
0
e−2xdx.
Answer: Z1
0
e−2xdx =1
2(1 −e−2)
Interactive Examples
Work the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/:
Section 6.8: Examples 1 through 5
Section 7.1: Example 4
†Lecture notes t o accompany Section 7.1 of Calc ulus by Hughes-Hallett et al.
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