The solution for x of the equation _√2^x dttt^2-1 = 2 is:

Question

The solution for $x$ of the equation $\displaystyle\int_{\sqrt2}^x \dfrac{dt}{t\sqrt{t^2-1}} = \dfrac{\pi}{2}$ is:

Accepted Answer

$$\int_{\sqrt 2 }^x {{{dt} \over {t\sqrt {{t^2} - 1} }}}  = {\pi  \over 2}$$

$$\therefore$$ $$\left[ {{{\sec }^{ - 1}}t} \right]_{\sqrt 2 }^x = {\pi  \over 2}$$

$$\left[ {} \right.$$ As $$\int {{{dx} \over {x\sqrt {{x^2} - 1} }}}  = {\sec ^{ - 1}}x$$ $$\left. {} \right]$$

$$ \Rightarrow {\sec ^{ - 1}}x - {\sec ^{ - 1}}\sqrt 2  = {\pi  \over 2}$$

$$ \Rightarrow {\sec ^{ - 1}}x - {\pi  \over 4} = {\pi  \over 2}$$

$$ \Rightarrow {\sec ^{ - 1}}x = {\pi  \over 2} + {\pi  \over 4}$$

$$ \Rightarrow {\sec ^{ - 1}}x = {{3\pi } \over 4}$$

$$ \Rightarrow x = \sec {{3\pi } \over 4}$$

$$ \Rightarrow x =  - \sqrt 2 $$

Solutions · 1