Take:
u = ln(2x+1)
du = 2/(2x+1) dx
dv = dx
v = x
|ln(2x+1) dx =
= xln(2x+1) - |2x/(2x+1) dx
= xln(2x+1) - |{1 - 1/(2x+1)} dx
= xln(2x+1) - |dx + |1/(2x+1) dx
= xln(2x+1) - x + (1/2)ln(2x+1) + const
The antiderivative of ln(2x+1) using integration by parts.
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