Question

Why do I get 2 different solution for same integral?

Answer 1

more info is required.

Answer 2

tell us the integral, tell us the method you used, tell us your answers....

The most likely direct answer toyour questions is
'because you made a mistake in one of them'

Answer 3

the steps down to the integrals are correct.

I have to admit that I used Wolfram to calculate the integrals, but it comes out with the answer
-1/4 ln(2x-1) for BOTH the variants as expected.

I suggest you check the method of integration for the right hand column.