Question

How do you integrate `int x^3sqrt(1+x^2)` using integration by parts?

Answer 1

`((1+x^2)^(3/2)(3x^2-2))/15+C`

Explanation:

`I=intx^3sqrt(1+x^2)dx`

Before trying any integration by parts, we should attempt more basic substitutions to simplify the integral. Let `u=1+x^2`. This implies that `du=2xdx` and `x^2=u-1`.

Rearranging the integral:

`I=1/2intx^2sqrt(1+x^2)(2xdx)=1/2int(u-1)sqrtudu`

So, we see that integration by parts won't be necessary at all!

`I=1/2int(u^(3/2)-u^(1/2))du=1/2(u^(5/2)/(5/2)-u^(3/2)/(3/2))`

`color(white)I=1/2(2/5u^(5/2)-2/3u^(3/2))=u^(5/2)/5-u^(3/2)/3=(3u^(5/2)-5u^(3/2))/15`

`color(white)I=(u^(3/2)(3u-5))/15=((1+x^2)^(3/2)(3x^2-2))/15+C`