Question

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

Answer 1

`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`