# Integrating the natural log

1. Feb 1, 2009

### Enujybab

1. The problem statement, all variables and given/known data
How would I begin to integrate ln(t+1) from 0 to e^2x?

2. Relevant equations
d/dx[log base a of u]=1/(lna)u du/dx

Can the original equation be manipulated to use this derivative?

3. The attempt at a solution
Not sure where to start.

2. Feb 1, 2009

### Dick

It's the sort of problem where you could actually guess the antiderivative, but if you can't, integrate by parts.

3. Feb 7, 2009

### masterslave

It's an integration by part question. First, use a substitution to get it to one variable instead of a polynomial in the logarithm.
w=t+1
dw=dt
Substitute (I'm going to revert the limits in the end to the original variable, so you know):
Integrate of ln(w)dw
Let:
u=ln(w) and dv=dw
du=(1/w)*dw and v=w

w*ln(w)-Int(w*(1/w)*dw)
w*ln(w)-Int(dw)
w*ln(w)-w
(t+1)*[ln(t+1)-1]