Where have I gone wrong in this integral by parts

[vande060]

Homework Statement

∫ ln(2x+1)dx

Homework Equations

The Attempt at a Solution

∫ ln(2x+1)dx

1/2∫2ln(2x+1)dx

t = 2x+1
dt = 2dx

1/2∫ln(t)dt

u = ln(t)
du = 1/t dt
dv = dt
v = t

tln(t) - ∫ t*1/t dt
tln(t) - ∫ dt
tln(t) - t

1/2*[(2x+1)ln(2x+1) - (2x +1)]

instead of this answer my book gives

1/2*(2x+1)ln(2x+1) - x

where did I go wrong?

 

[th13teenj]

Try it without the initial substitution and you get the answer in the book.

I couldn't find anything wrong in your steps. I suppose if you expand your answer, you get you get the books answer + 1/2. I'm not sure, but i guess that can just be absorbed into the constant c. The easiest way to realize this is to just take the derivative.

 

[dextercioby]

You didn't go wrong, except that you forgot the integration constant at the end. If you had put it, you would have seen why the answer you found and the one in the book difer by 1/2.