Question:
Evaluate the following integral:
$\int \frac{x^{2}+1}{x^{4}+7 x^{2}+1} d x$
Solution:
re-writing the given equation as
$\int \frac{1+\frac{1}{x^{2}}}{x^{2}+7+\frac{1}{x^{2}}} d x$
$\int \frac{1+\frac{1}{x^{2}}}{\left(x-\frac{1}{x}\right)^{2}+9} d x$
Assume $t=x-\frac{1}{x}$
$\mathrm{dt}=\left(1+\frac{1}{\mathrm{x}^{2}}\right) \mathrm{d} \mathrm{x}$
$\int \frac{d t}{(t)^{2}+9}$
Using identity $\int \frac{1}{x^{2}+1} d x=\arctan (x)$
$\frac{1}{3} \arctan \left(\frac{t}{3}\right)+c$
Substituting $\mathrm{t}$ as $\mathrm{x}-\frac{1}{\mathrm{x}}$
$\frac{1}{3} \arctan \left(\frac{\mathrm{x}-\frac{1}{\mathrm{x}}}{3}\right)+\mathrm{c}$
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