Find the integral: (x-1 x )^ 2 d x - Vidyadip

Question

Find the integral: $\int\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)^{2} d x$

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Answer

$\int\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)^{2} d x$
= $\int\left(x+\frac{1}{x}-2\right) d x$
= $\int x d x+\int \frac{1}{x} d x-2 \int 1 d x$
Now we know that,
$^{\int x^{n} d x}=\frac{x^{n+1}}{n+1}+c$
Therefore,
$\int x d x+\int \frac{1}{x} d x-2 \int 1 d x$ = $\frac{x^{2}}{2}+\log |x|-2 x+C$

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