Two different solutions of the same integral

Question

Considering $$\int\frac{\ln(x+1)}{2(x+1)}dx$$ I first solved it seeing it similar to the derivative of $\ln^2(x+1)$ so multiplying by $\frac22$ the solution is $$\int\frac{\ln(x+1)}{2(x+1)}dx=\frac{\ln^2(x+1)}{4}+const.$$. But then we can solve it using by parts' method and so this is the solution that I found: $$\frac12\int\frac{\ln(x+1)}{(x+1)}dx=\frac12\ln(x+1)\ln(x+1)-\frac12\int\frac{\ln(x+1)}{x+1}dx$$ Seeing it as an equation I brought the integral $-\frac12\int\frac{\ln(x+1)}{x+1}dx$ to the left so that I obtain $$\int\frac{\ln(x+1)}{(x+1)}dx=\frac12\ln(x+1)\ln(x+1)+const.$$ so $$\int\frac{\ln(x+1)}{(x+1)}dx=\frac12\ln^2(x+1)+const.$$. I know that the first solution is correct but I used to way of solution that seem to be correct. How is it possible? Where is the mistake? Thank you in advance for your help!

Answer 1

Both answers are the same. Can you see that

$\int\frac{ln(x+1)}{(x+1)}dx=\frac{1}{2}\ln(x+1)ln(x+1)+cost.$

$\frac{1}{2}\int\frac{ln(x+1)}{(x+1)}dx=\frac{1}{4}\ln(x+1)ln(x+1)+cost.$

Answer 2

Divide the last step by $2$ to get the desired answer, as $2*constant=constant$.

Answer 3

Look at the denominator in the integrand of your final result. The 2 is missing that was present in the original integral. Divide both sides of the equation by 2 and you get the desired result.