Q

# Stuck here, help me understand: - For(the set of natural numbers), the integral is equal to: (where c is a constant of integration) - Integral Calculus - JEE Main

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)

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

Views

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)

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

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