Why I'm getting wrong answer? What is wrong in my solution?

Question

I'm learning single variable calculus right now. Right now trying to understand integration with partial fraction. I'm confused in a problem from sometime. I think I'm doing right but answer in my book is something else. Please have a look at the images.

The answer given in my book is $\frac{x}{2}+ \log x - \frac{3}{4}\log(1-2x) + C$
Please help. thank you in advance.

Answer 1

Actually $\ln2x=\ln2+\ln x$

So, $\ln2x+C=\ln x+(\ln2+C)=\ln x+C'$

Using de facto partial fraction decomposition,

$$\dfrac{1-x^2}{x(1-2x)}=A+\dfrac Bx+\dfrac C{1-2x}$$

$$\iff1-x^2=Ax(1-2x)+B(1-2x)+Cx=-2Ax^2+x(A+C-2B)+B$$

Comparing the constants $B=1$

Comparing the coefficients of $x^2,-1=-2A$

Compare the coefficients of $x$

Answer 2

$\log2x=\log2+\log x$ and $\log2$ is absorbed in $C$. So your answer agrees with what is in the book.