What's the integral of x*ln(x)^2?

The integral of x*ln(x)^2 is (x^2/2)*ln(x)^2 - (x^2/4)*ln(x) + (x^2/8) + C.

To solve this integral, we can use integration by parts. Let u = ln(x)^2 and dv = x dx. Then du/dx = 2ln(x)/x and v = x^2/2. Using the formula for integration by parts, we have:

∫x*ln(x)^2 dx = uv - ∫v du/dx dx
= (x^2/2)*ln(x)^2 - ∫(x^2/2)*(2ln(x)/x) dx
= (x^2/2)*ln(x)^2 - x^2*ln(x)/4 + ∫x dx/4
= (x^2/2)*ln(x)^2 - (x^2/4)*ln(x) + (x^2/8) + C

Therefore, the integral of x*ln(x)^2 is (x^2/2)*ln(x)^2 - (x^2/4)*ln(x) + (x^2/8) + C.