Question

How do you use substitution to integrate `(r^2) √(r^(3) +3)dr`?

Answer 1

`intr^2/sqrt(r^3+3)dr=2/3sqrt(r^3+3)+C`

Explanation:

So, we want to determine

`intr^2/sqrt(r^3+3)dr`

The substitution should be picked in such a way that the differential of what we've picked will show up in the integral.

The best choice is `u=r^3+3`, as

`du=3r^2dr`

`3r^2dr` does not show up in this exact form in the integral; however, `1/3du=r^2dr` shows up in the numerator of the integral. Thus, we can substitute and now have

`1/3int(du)/sqrt(u)=1/3intu^(-1/2)du` (We've factored `1/3` outside of the integral)

Integrate:

`(1/3)(2)sqrt(u)+C`

Rewriting in terms of `x` yields

`intr^2/sqrt(r^3+3)dr=2/3sqrt(r^3+3)+C`