6. Integration
6.2 Substitution
To learn one of many powerful methods to computing more complicated indefinite integrals. Substitution method will often simplify the calculation.
Example 1.
Getting acquainted: Find the following indefinite integral
Z x
5
( x
6
+ 1)
3 d x
How
Based on the reverse chain rule.
Recall of chain rule:
Let u = g ( x ) = x
6
+ 1. Then, the differential of u d u =
So, d x =
Thus,
Z x
5
( x
6
+ 1)
3 d x
Check (by differentiation as we did in 6.1):
1

What
The Substitution Rule: Let u = g ( x ), we have
Z f ( g ( x )) g
0
( x ) d x =
Z f ( u ) d u, 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 x ( x
2
+ 1)
2015 d x
Example 3.
Z x
√
2 + x 2 d x
2

Example 4.
Z x − 7
( x 2 − 14 x ) 2 d x
Example 5.
Z e x e x + 5 d x
Example 6.
Z xe x
2 d x
3

Example 7.
Z (ln x )
2 d x x
Example 8.
Z e x
4
+ e x
(4 x
3
+ e x
) d x,
Example 9.
Z x (1 − x
2
)
3 / 2 d x
4