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  4. The solution of ∫√2x (dt/t√t2-1)=(π/12) is

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Q. The solution of $\int_{\sqrt{2}}^{x} \frac{dt}{t\sqrt{t^{2}-1}}=\frac{\pi}{12}$ is

KEAMKEAM 2013Integrals

Solution:

Given,
$\int_{\sqrt{2}}^{x} \frac{d t}{t \sqrt{t^{2}-1}}=\frac{\pi}{12}$
$ \Rightarrow \left[\sec ^{-1} t\right]_{\sqrt{2}}^{x}=\frac{\pi}{12}$
$\Rightarrow \, \sec ^{-1} x-\sec ^{-1}(\sqrt{2})=\frac{\pi}{12}$
$\Rightarrow \, \sec ^{-1} \times \frac{\pi}{4}=\frac{\pi}{12}$
$\Rightarrow \, \sec ^{-1} x=\frac{\pi}{12}+\frac{\pi}{4}=\frac{4 \pi}{12}=\frac{\pi}{3}$
$\therefore \, x=\sec \frac{\pi}{3}=2$