What is the antiderivative of ln(x+1)?

1 Answer
Jun 2, 2016

(x+1)ln(x+1)-x+C

Explanation:

We have

I=intln(x+1)dx

We will use integration by parts:

intudv=uv-intvdu

For intln(x+1)dx, let u=ln(x+1) and dv=dx. These imply that du=1/(x+1)dx and v=x.

Thus,

I=xln(x+1)-intx/(x+1)dx

Solving the remaining integral:

intx/(x+1)dx=int(x+1-1)/(x+1)dx=int1-1/(x+1)dx

=x-ln(x+1)+C

So,

I=xln(x+1)-(x-ln(x+1))+C

I=(x+1)ln(x+1)-x+C