Integrate the function: x-1 x^ 2 -1 - Vidyadip

Question

Integrate the function: $\frac{x-1}{\sqrt{x^{2}-1}}$

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Answer

Here, $\int \frac{x-1}{\sqrt{x^{2}-1}} d x=\int \frac{x}{\sqrt{x^{2}-1}} d x-\int \frac{1}{\sqrt{x^{2}-1}} d x$
For $\int \frac{x}{\sqrt{x^{2}-1}} d x$ , Let $x^2 - 1 = t$
$\Rightarrow 2x\ dx = dt$
$\Rightarrow \int \frac{\mathrm{x}}{\sqrt{\mathrm{x}^{2}-1}} \mathrm{dx}=\frac{1}{2} \int \frac{\mathrm{dt}}{\sqrt{\mathrm{t}}}$
$= \frac{1}{2} \int t^{-\frac{1}{2}} d t$
$= \frac{1}{2}\left[2 t^{\frac{1}{2}}\right]=\sqrt{t}=\sqrt{x^{2}-1}$
$\Rightarrow \int \frac{\mathrm{x}-1}{\sqrt{\mathrm{x}^{2}-1}} \mathrm{dx}=\int \frac{\mathrm{x}}{\sqrt{\mathrm{x}^{2}-1}} \mathrm{dx}-\int \frac{1}{\sqrt{\mathrm{x}^{2}-1}} \mathrm{d} \mathrm{x}$
$= \sqrt{x^{2}-1}- \log |x+\sqrt{x^{2}-1}|+C~~~$

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