Question

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

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

Answer 1

Answer: (D)

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