What's the integral of x*sqrt(x^2+1)?

The integral of x*sqrt(x^2+1) is (x^2+1)^(3/2)/3 + C.

To solve this integral, we can use the substitution u = x^2 + 1. Then, du/dx = 2x and dx = du/2x. Substituting these into the integral, we get:

∫x*sqrt(x^2+1) dx = ∫sqrt(u) du/2 = 1/2 ∫u^(1/2) du

Integrating u^(1/2), we get:

1/2 ∫u^(1/2) du = 1/2 * (2/3) u^(3/2) + C = (u^(3/2))/3 + C

Substituting back u = x^2 + 1, we get:

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

Therefore, the integral of x*sqrt(x^2+1) is (x^2+1)^(3/2)/3 + C.