QJEEM24Ma2f7b: JEE Main 2024 Mathematics - Class 12 JEEM24Ma2f7b Solution: For x ∈ ( − π 2 , π 2 ) , if y ( x ) = ∫ cosec x + sin ⁡ x…

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QJEEM24Ma2f7b: JEE Main 2024 Mathematics - Class 12 JEEM24Ma2f7b Solution: For x ∈ ( − π 2 , π 2 ) , if y ( x ) = ∫ cosec x + sin ⁡ x…

For $x \in (- \frac{π}{2}, \frac{π}{2})$ , if $y (x) = \int \frac{cosec x + \sin x}{cosec x \sec x + \tan x \sin^{2} x} d x$ , and $lim_{x \to {(\frac{π}{2})}^{-}} y (x) = 0$ then $y (\frac{π}{4})$ is equal to

Multiple Choice Options

A −12tan−1⁡(12)
B tan−1⁡(12)
C 12tan−1⁡(12)
D 12tan−1⁡(−12)

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Accepted Answer

$\begin{aligned} y (x) = \int \frac{(1 + \sin^{2} x) \cos x}{1 + \sin^{4} x} d x \\ Put \sin x = t \\ = \int \frac{1 + t^{2}}{t^{4} + 1} d t = \frac{1}{\sqrt{2}} \tan^{- 1} \frac{(t - \frac{1}{t})}{\sqrt{2}} + C \\ x = \frac{π}{2}, t = 1 ∴ C = 0 \\ y (\frac{π}{4}) = \frac{1}{\sqrt{2}} \tan^{- 1} (- \frac{1}{2}) \end{aligned}$

QJEEM24Ma2f7b: JEE Main 2024 Mathematics - Class 12 JEEM24Ma2f7b Solution: For x ∈ ( − π 2 , π 2 ) , if y ( x ) = ∫ cosec x + sin ⁡ x…

Multiple Choice Options

Step-by-Step Verified Solution

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