Question

How do you integrate `x( (x^2) +1 )^ (1/2)`?

Answer 1

`1/3(x^2+1)^(3/2)+C`

Explanation:

`intx(x^2+1)^(1/2)dx`

Let `u=x^2+1`. Differentiating this shows that `du=2xcolor(white).dx`.

We already have `xcolor(white).dx` in the integrand, so all we need to do is multiply the integrand by `2`. To balance this out, multiply the exterior of the integral by `1//2`.

`=1/2intunderbrace((x^2+1)^(1/2))_(u^(1/2))overbrace((2xcolor(white).dx))^(du)`

`=1/2intu^(1/2)du`

`=1/2u^(3/2)/(3/2)`

`=1/3u^(3/2)`

`=1/3(x^2+1)^(3/2)+C`