Question

Solve:
${\sin }^{-1}\left(\frac{2^{x+1}}{1+4^x}\right)$

Answer 1

Consider the following differential eq.

$y={\sin }^{-1}\left(\frac{2^{x+1}}{1+4^x}\right)$

Let $2^x=\tan t$

$={\sin }^{-1}\left(\frac{2\tan t}{1+{\tan }^{2}t}\right)$

$={\sin }^{-1}\left(\sin 2t\right)$.

$=2t$ ...... (1)

Put the value of t in above eq(1) we get.

$=2{\tan }^{-1}\left(2^x\right)$

Hence, this is the required answer.