How to integrate x^2*sqrt(x+1)?

To integrate x^2*sqrt(x+1), use substitution with u = x+1 and then integration by parts.

First, substitute u = x+1, so that x = u-1 and dx = du. Then, the integral becomes:

∫(u-1)^2*sqrt(u) du

Expand the square and simplify:

∫(u^2 - 2u + 1)*sqrt(u) du
= ∫u^(5/2) - 2u^(3/2) + u^(1/2) du
= (2/7)u^(7/2) - (4/5)u^(5/2) + (2/3)u^(3/2) + C

Substitute back u = x+1 to get the final answer:

(2/7)(x+1)^(7/2) - (4/5)(x+1)^(5/2) + (2/3)(x+1)^(3/2) + C