JEE Advanced Indefinite Integration 2

Calculus Level 5

If I = ∫ d x sin ⁡ x sin ⁡ ( 2 x + α ) \displaystyle I=\int \dfrac{dx}{\sin x \sqrt{\sin (2x+\alpha)}} for 0 < α < π 2 0<\alpha<\dfrac{\pi}{2} and f ( x ) = log ⁡ ∣ cot ⁡ x + cot ⁡ α + cot ⁡ 2 x − csc ⁡ 2 α ∣ f(x)=\log \left |\cot x+\cot \alpha + \sqrt{\cot^2x-\csc^2\alpha }\right | , then find I I .

This problem is a part of My picks for JEE Advanced 1
None of the others 1 2 f ( x ) + C \dfrac{1}{2}f(x)+C f ( x ) sin ⁡ α + C \dfrac{f(x)}{\sqrt{\sin\alpha}}+C f ( x ) cos ⁡ α + C \dfrac{f(x)}{\sqrt{\cos\alpha}}+C

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Pranjal Jain by Pranjal Jain , 23, India

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1 solution

I can be determined in a closed form, but let's just observe and plug in x = π / 4 x=\pi/4 and then we see that derivative of I gives a real number where f(x) doesn't.

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