∫x∞0s+nx2−13xdx By using by parts =cos⁡n∫xx2−132dx−∫ddn(cos⁡n)∫xx2−132dndx=cos⁡x2x2−1−32+1−32+1−∫1n×12x2−1−32+1−32+1dx=−cos⁡xx∗−11x+∫1xx2−1dx=−cos⁡nn2−1+sec−1⁡(x)+c

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$\int \frac{x . \log x}{{(\sqrt{x^{2} - 1})}^{3}} d x =$

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$\sec^{- 1} x + \frac{\log x}{\sqrt{x^{2} - 1}} + c$

$\sec^{- 1} x - \frac{\log x}{\sqrt{x^{2} - 1}} + c$

$\frac{\log x}{\sqrt{x^{2} - 1}} - \sec^{- 1} x + c$

$\frac{- \log x}{\sqrt{x^{2} - 1}} - \sec^{- 1} x + c$

answer is B.

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Detailed Solution

$\int \frac{x \infty 0 s + n}{{(x^{2} - 1)}^{\frac{3}{x}}} d x$
$By using by parts$
$\begin{aligned} = \cos n \int \frac{x}{{(x^{2} - 1)}^{\frac{3}{2}}} d x - \int (\frac{d}{d n} (\cos n) \int \frac{x}{{(x^{2} - 1)}^{\frac{3}{2}}} d n) d x \\ = \frac{\cos x}{2} \frac{{(x^{2} - 1)}^{- \frac{3}{2} + 1}}{- \frac{3}{2} + 1} - \int \frac{1}{n} \times \frac{1}{2} \frac{{(x^{2} - 1)}^{- \frac{3}{2} + 1}}{- \frac{3}{2} + 1} d x \\ = \frac{- \cos x}{{(x^{*} - 1)}^{\frac{1}{x}}} + \int \frac{1}{x \sqrt{x^{2} - 1}} d x \\ = \frac{- \cos n}{\sqrt{n^{2} - 1}} + \sec^{- 1} (x) + c \end{aligned}$