Question:
Evaluate the following integral:
$\int \frac{x^{2}+1}{x^{4}-x^{2}+1} d x$
Solution:
re-writing the given equation as
$\int \frac{1+\frac{1}{x^{2}}}{x^{2}-1+\frac{1}{x^{2}}} d x$
$\int \frac{1+\frac{1}{x^{2}}}{\left(x-\frac{1}{x}\right)^{2}+1} d x$
Substituting $t$ as $x-\frac{1}{x}$
$\left(1+\frac{1}{x^{2}}\right) d x=d t$
$\int \frac{d t}{t^{2}+1}$
Using identity $\int \frac{1}{\mathrm{x}^{2}+1} \mathrm{dx}=\arctan (\mathrm{x})$
$\arctan t+c$
Substituting tas $\mathrm{x}-\frac{1}{\mathrm{x}}$
$\arctan \left(x-\frac{1}{x}\right)+c$
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