Mathematics
ASSERTION

If $$\displaystyle \int \frac{2^{x}}{\sqrt{1-4^{x}}}dx=k\sin^{-1}\left ( 2^{x} \right )+C,$$ then the value of $$\displaystyle k=\frac{1}{\log 2}$$

REASON

$$\displaystyle \int \frac{dx}{\sqrt{1-x^{2}}}=\sin^{-1}\left ( x \right )+C$$


ANSWER

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion


SOLUTION
$$\displaystyle \int \frac{2^{x}}{\sqrt{1-4^{x}}}dx=\int \frac{2^{x}}{\sqrt{1-\left ( 2^{x} \right )^{2}}}dx$$

   $$\displaystyle =\frac{1}{\log2}\sin^{-1}\left ( \frac{2^{x}}{1} \right )+C$$

Substitute $$2^{x}=t$$, $$2^{x}\log 2dx=dt$$

   $$\displaystyle =k\sin^{-1}\left ( \frac{2^{x}}{1} \right )+C$$

  $$\therefore $$   $$\displaystyle k=\frac{1}{\log 2}$$
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Assertion & Reason Medium Published on 17th 09, 2020
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