Question

Evaluate the integral of x*ln(1+x) dx by first making a substitution and then using integration by parts.



I let u=1+x and du = dx but then ln(u) du can't integrate to anything?

Answer 1

you need to get rid of that pesky ln(1+x). So,

u = ln(1+x)
du = 1/(1+x) dx
dv = x dx
v = 1/2 x^2

∫u dv = uv - ∫v du
= 1/2 x^2 ln(1+x) - 1/2 ∫x^2/(1+x) dx

which is not so hard, eh?