Mathematics

The solution for x of the equation $$\displaystyle\int^x_{\sqrt{2}}\dfrac{dt}{t\sqrt{t^2-1}}=\dfrac{\pi}{2}$$ is?


ANSWER

$$- \sqrt2$$


SOLUTION
$$\displaystyle \int_{\sqrt 2}^{x} \dfrac {dt}{t\sqrt {t^2 -1}}=\dfrac {\pi}{2}$$
Let $$t=\sec \theta$$
$$dt=\sec \theta \tan \theta \ d\theta$$
$$\displaystyle \int_{\pi /4}^{\sec^{-1}x} \dfrac {\sec \theta \tan \theta d\theta}{\sec \theta \sqrt {\sec^2 \theta -1}} \ \Rightarrow \displaystyle \int_{\pi /4}^{\sec^{-1}\theta} \dfrac {\sec \theta \tan \theta d\theta }{\sec \theta \tan \theta }=\dfrac {\pi}{2}$$
$$\displaystyle \int_{\pi /4}^{\sec^{-1}x} d\theta =\dfrac {\pi}{2}$$
$$\sec^{-1}x-\dfrac {\pi}{4}=\dfrac {\pi}{2}$$
$$\sec^{-1}x=\dfrac {\pi}{2}+\dfrac {\pi}{4}\ \Rightarrow x=\sec \left (\dfrac {3\pi}{4}\right)$$
$$\boxed {x=-\sqrt 2}$$
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Single Correct Medium Published on 17th 09, 2020
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