Question

How do you integrate `int x^3*sqrt(4+x^2)` using integration by parts?

Answer 1

not by using integration by part

`u = x^2+4`
`du = 2x`

`1/2int2x^3*sqrt(x^2+4) du`

`1/2intx^2*sqrt(u)du`

`x^2 = u-4`

`1/2int(u-4)*sqrt(u)du`

expand

`1/2intu*sqrt(u)-4sqrt(u)du`

`1/2intu^(3/2)-4sqrt(u)du`

`1/2[2/5u^(5/2)-8/3u^(3/2)]+C`

`[1/5u^(5/2)-4/3u^(3/2)]+C`