Go To Your Study Dashboard

Get Answers to all your Questions

header-bg

For $x^2 \neq n\pi +1, n\in N$ (the set of natural numbers), the integral

$\int x\sqrt{\frac{2\sin(x^2-1) - \sin2(x^2 -1)}{2\sin(x^2-1) + \sin2(x^2 -1)}}dx$ is equal to:

(where c is a constant of integration)
Option 1)

$\log_e |\frac{1}{2}\sec^2(x^2 -1)| +c$
Option 2)
$\frac{1}{2}\log_e |\sec^2(x^2 -1)| +c$
Option 3)
$\frac{1}{2}\log_e \left |\sec^2\left (\frac{x^2 -1}{2} \right )\right | +c$
Option 4)
$\log_e \left |\sec^2\left (\frac{x^2 -1}{2} \right )\right | +c$

Answers (1)

best_answer

Integral of Trigonometric functions -

$\int \tan x\: dx= \ln \left | \sec x \right |+C$

$\int \cot x\: dx= \ln \left | \sin x \right |+C$

$\int \sec x\: dx= \ln \left | \sec x+\tan x \right |+C$

$\int cosec \: x\: dx= \ln \left | cosec \: x- \cot x \right |+C$

-

Put $\left ( x^{2} -1\right )=t;\; \; \; \; 2xdx=dt$

$\int x\sqrt{\frac{2\sin \left ( x^{2}-1 \right )-\sin 2\left ( x^{2}-1 \right )}{2\sin \left ( x^{2}-1 \right )+\sin 2\left ( x^{2}-1 \right )}}\: \; \; dx=\frac{1}{2}\int \sqrt{\frac{2\sin \left ( t \right )-\sin 2t}{2\sin \left ( t \right )+\sin \left ( 2t \right )}}\: \: dt$

$\because \sin \left ( 2t \right )=2\sin \left ( t \right )\cos \left ( t \right )$

$\Rightarrow \frac{1}{2}\int \sqrt{\frac{1-\cos \left ( t \right )}{1+\cos \left ( t \right )}}\: \: dt\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \because \cos \left ( 2t \right )=1-2\sin ^{2}\left ( t \right )$

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

$\Rightarrow \frac{1}{2}\int \tan \left ( \frac{+}{2} \right )dt$

$\Rightarrow \ln \left | \sec \left ( \frac{+}{2} \right ) \right | +C$

replace t with $\left ( x^{2}-1 \right )$

$\Rightarrow \ln \left | \sec \left ( \frac{x^{2}-1}{2} \right ) \right |+C$

Option 1)

$\log_e |\frac{1}{2}\sec^2(x^2 -1)| +c$

Option 2)

$\frac{1}{2}\log_e |\sec^2(x^2 -1)| +c$

Option 3)

$\frac{1}{2}\log_e \left |\sec^2\left (\frac{x^2 -1}{2} \right )\right | +c$

Option 4)
$\log_e \left |\sec^2\left (\frac{x^2 -1}{2} \right )\right | +c$

Posted by

admin

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

Trending Articles/News

anam-khan

JEE Main Result 2024 Live: JEE Mains session 2 results at jeemain.nta.ac.in soon; rank predictor

anam-khan

JEE Main 2024 session 2 final answer key out at jeemain.nta.ac.in; steps to check probable scores

anam-khan

JEE Main Result 2024 Live: Download session 2 final answer key at jeemain.nta.ac.in; rank predictor, cutoff

Latest Question