Question

How do you use substitution to integrate `(2x(x^2 + 1)^23)dx`?

Answer 1

`int[2x(x^2+1)^23]dx=1/24(x^2+1)^24+C`

Explanation:

Use the substitution : Let `u=x^2+1`

Then `du=2xdx` so this can be plugged into the integral. It may help to rewrite it so `2x` and `dx` are next to one another:

`int(x^2+1)^23(2xdx)`

We may hence write the original integral in terms of x into a new equivalent integral in terms of u as follows :

`intu^23du=u^24/24+C`

We now substitute back in terms of x to obtain the final answer as `(x^2+1)^24/24+C`