Question

can someone help me integrate x/(2x-1)

Answer 1

u must integrate by part.

u = x
dv = dx/(2x-1)

if you have specific questions let me know

Answer 2

Or, long division has been working nicely for generations.

\(\dfrac{x}{2x-1} = \dfrac{1}{2} + \dfrac{1/2}{2x-1}\)

Answer 3

Or a substitution! Lots of ways to go about it.

Let \(u=2x-1~~\Rightarrow~~\dfrac{1}{2}du=dx\), and you also have \(x=\dfrac{u+1}{2}\).

\[\begin{align*}\int\frac{x}{2x-1}~dx&=\frac{1}{2}\int\frac{\frac{u+1}{2}}{u}~du\\
&=\frac{1}{4}\int\frac{u+1}{u}~du\\
&=\frac{1}{4}\int\left(1+\frac{1}{u}\right)~du\end{align*}\]