Evaluate the following integral:


Evaluate the following integral:

$\int \frac{x^{3}-1}{x^{3}+x} d x$


$I=\int \frac{x^{3}-1}{x^{3}+x} d x=\int 1-\frac{x+1}{x^{3}+x} d x$

$=\int 1 \mathrm{dx}-\int \frac{\mathrm{x}+1}{\mathrm{x}^{3}+\mathrm{x}} \mathrm{dx}$


$x+1=A\left(x^{2}+1\right)+(B x+C)(x)$

Equating constants


Equating coefficients of $x$


Equating coefficients of $x^{2}$



$I=-\int \frac{d x}{x}-\int \frac{-x+1 d x}{x^{2}+1}+\int d x$

$I=-\int \frac{d x}{x}+\int \frac{x d x}{x^{2}+1}-\int \frac{d x}{x^{2}+1}+\int d x$

$=-\log |\mathrm{x}|+\frac{1}{2} \log \left|\mathrm{x}^{2}+1\right|-\tan ^{-1} \mathrm{x}+\mathrm{x}+\mathrm{c}$

$I=x-\log |x|+\frac{1}{2} \log \left|x^{2}+1\right|-\tan ^{-1} x+c$

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