6. Integration
6.2 Substitution
Why
Example 1. Getting acquainted: Find the following indefinite integral
Z
x5 (x6 + 1)8 dx
How
Based on the reverse chain rule.
Recall of chain rule:
Let u = g(x) = x6 + 1. Then, the differential of u
du =
So,
dx =
Thus,
Z
x5 (x6 + 1)8 dx
1
Check (by differentiation as we did in 6.1):
What
The Substitution Rule: Let u = g(x), we have
Z
Z
0
f (g(x))g (x) dx = f (u) du,
assuming that u = g(x) is a continuously differentiable function whose range is an interval on which f
is continuous.
Trick
Choosing the right u in the integrand. It depends on the integral. Practice for gaining experience.
SOME PRACTICE. Finding the following indefinite integrals.
Example 2.
Z
(4x − 5)6 dx
Example 3.
Z
x(−x2 + 9)2014 dx
2
Example 4.
Z
√
x 2 + 5x2 dx
Example 5.
x − ln2
dx
− 2xln2
Z
x2
Example 6.
Z
e2x
dx
e2x + e
3
Example 7.
Z
4
12x3 ex dx
Example 8.
Z
x3
dx
(x4 + 2)3
Example 9.
Z
2 + 5e4−3x dx
4
Example 10.
Z
(lnx − 1)5
dx
x
Example 11.
Z
8e3/x x2 dx
5
Example 12. Find f (x) if f (5) = 0 and f 0 (x) =
x
x2 +3
More examples
6