Question

How do you integrate `int sqrtxsqrt(xsqrtx+1)` using substitution?

Answer 1

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

Explanation:

This integral is a power integral. The square root operator helps to conceal de true problem.

`sqrtxsqrt(xsqrtx+1) = x^(1/2)(x^(3/2)-1)^(1/2)=(2/3)^2d/dx(x^(3/2)-1)^(3/2)`

So

`int sqrtx(sqrt(xsqrtx+1))dx = (2/3)^2(x^(3/2)-1)^(3/2) + C`