Question

How do you integrate `int xsqrt(2x+1)dx`?

Answer 1

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

Explanation:

Let `2x+1=t^2 :. x=(t^2-1)/2, and, dx=tdt.`

Therefore, the Reqd. Integral`=int{((t^2-1)/2)(sqrtt^2)(t)}dt`

`=1/2int(t^4-t^2)dt=1/2(t^5/5-t^3/3)=t^3/30(3t^2-5)`

`=1/30(t^2)^(3/2)(3t^2-5)`

`=1/30(2x+1)^(3/2){3(2x+1)-5}`

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