Help with a calculus problem (integration by parts)

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Help with a calculus problem (integration by parts)

Mathematics
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\[integrate \ln(2x+1)dx\]
I have some work, give me a few minutes to input it
\[\int\limits_{}^{}\ln(2x+1)dx = x*\ln(2x+1) - \int\limits_{}^{}\frac{ 2x }{ 2x +1 }\]

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I don't know how to take the integral of 2x / 2x + 1. I've viewed solutions online and I've seen...
I don't know why they split it into (2x - 1 + 1)/2x +1 Everything else I can do, except I just don't understand that part
do you still remember how to do "long division" ?
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lol, i'm not gonna lie, it's been a while. but i can do it! thanks for answering. The way I initially approached it was by doing \[2\int\limits_{}^{}x * \frac{ 1 }{ 2x +1 }\] and then using \[uv - \int\limits_{}^{}vdu\] to solve. so maybe that's why i didn't initially think of doing long division
that gives you back the original integrand haha!
yeah that's why i was like what am i doing wrong?? haha
ikr! review this quick for long division https://www.khanacademy.org/math/algebra2/polynomial_and_rational/dividing_polynomials/v/polynomial-division
thanks so much for your help! i'll check the link out

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