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

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

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
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Single Correct Medium
$\lim _{ n\rightarrow \infty }{ \cfrac { 1 }{ \sqrt { n } \sqrt { n+1 } } +\cfrac { 1 }{ \sqrt { n } \sqrt { n+2 } } +....+\cfrac { 1 }{ \sqrt { n } \sqrt { 4n } } }$ is equal to
• A. $2$
• B. $4$
• C. $\sqrt { 2 } -1$
• D. $2\left( \sqrt { 5 } -1 \right)$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
The value of $\displaystyle\int\limits_{1}^{e^2}\dfrac{dx}{x}$
• A. $1$
• B. $-1$
• C. $-2$
• D. $2$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
$\int \dfrac {x^{2}dx}{(x\sin x + \cos x)^{2}}$ is equal to
• A. $\dfrac {\sin x + \cos x}{x\sin x + \cos x}$
• B. $\dfrac {x\sin x - \cos x}{x\sin x + \cos x} + c$
• C. $\dfrac {\sin x - \cos x}{x\sin x + \cos x}x$
• D. None of these

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\displaystyle\int \frac{1}{\left [ \left ( 1-x^{2} \right )\left \{ \left ( 2\sin ^{-1}x \right )^{2} -9\right \}\right ]^{1/2}}dx.$
• A. $\displaystyle \log \left [ 2\sin ^{-1}x+\sqrt{\left ( 2\sin ^{-1}x \right )^{2}-9} \right ].$
• B. $\displaystyle \frac{1}{2}\log \left [ 2\sin ^{-1}x+\sqrt{\left ( 2\sin ^{-1}x \right )^{2}+9} \right ].$
• C. $\displaystyle \frac{1}{2}\log \left [ 2\sin ^{-1}x+\sqrt{\left ( 2\cos ^{-1}x \right )^{2}-9} \right ].$
• D. $\displaystyle \frac{1}{2}\log \left [ 2\sin ^{-1}x+\sqrt{\left ( 2\sin ^{-1}x \right )^{2}-9} \right ].$

$\displaystyle\int { \cfrac { \log { x } }{ x } } dx$