While integrating a rational function I stumbled upon the following problem (In the calculation of the integral the substitution [tex]u=x+1, du=d(x+1)=dx[/tex] was used).(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\begin{align}

\int \frac{x^2}{(1+x)^2}\,dx &= \int \frac{(u-1)^2}{u^2}\,du

\\

&= \int du+\int \frac{du}{u^2}\ -2 \int \frac{du}{u}\

\\

&= u-\frac{1}{u}- 2 \log(u)

\\

&=1+x - \frac{1}{1+x}-2 \log(1+x)

\end{align}

[/tex]

The problem now is that if I substitute x back into the integral during step (2) I get [tex]x - \frac{1}{1+x}-2 \log(1+x)[/tex].

Obviously taking the derivative of both primitives yields the same integrand.

My problem with this is that instead of getting an unknown constant I get this unwanted extra 1. Secondly if I plug this integral into mathematica it gives the result without the constant 1.

So my question is why do I get different functions without having specified the integration constant?

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# Unwanted constants after integration

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