Question

If $\int \frac{2^x}{\sqrt{1-4^x}}dx=K{\sin }^{-1}\left(2^x\right)+C$, then $K$ is equal to

• A. $\ell n2$
• B. $\frac{1}{2}\ell n2$
• C. $\frac{1}{2}$
• D. $\frac{1}{\ell n2}$

Answer 1

Answer: (D)

Let $2^x=t{2}^x\log 2dx=dt$
$I=\frac{1}{\log 2}\int \frac{dt}{\sqrt{1-t^2}}=\frac{1}{\ln 2}{\sin }^{-1}(t)+c=\frac{1}{\ln 2}{\sin }^{-1}\left(2^x\right)+c$
$\Rightarrow k=\frac{1}{\ln 2}$