How do you integrate by substitution #int x^2(x^3+5)^4 dx#?

1 Answer
Oct 31, 2016

# int x^2(x^3+5)^4 dx =(x^3+5 )^5/15+C #

Explanation:

We want to find # I=int x^2(x^3+5)^4 dx #

Let # u=x^3+5 => (du)/dx=3x^2 #, or #3x^2dx/(du)=1#

We can then rewrite #I# as follows:

# I=int x^2u^4 dx #
# :. I=1/3 int u^4 (3x^2)dx #
# :. I=1/3 int u^4 (3x^2)dx/(du)du # (by the chain rule)

And using the above result we can now substitute to get:
# I=1/3 int u^4 (1) du #
# :. I=1/3 int u^4 du #
# :. I=1/3 u^5/5+C #
# :. I=(x^3+5 )^5/15+C #