Question

Evaluate: \[ \int \tan^{-1}\left( \sqrt{\frac{1-\sin x}{1+\sin x}} \right)\,dx \]

• A. \(\dfrac{\pi x}{2}-\dfrac{x^2}{4}+C\)
• B. \(\dfrac{\pi x}{4}-\dfrac{x^2}{2}+C\)
• C. \(\dfrac{\pi x}{4}-\dfrac{x}{4}+C\)
• D. \(\dfrac{\pi x}{4}-\dfrac{x^2}{4}+C\)

Hint:

Remember the identity: \[ \tan\left(\frac{\pi}{4}-\frac{x}{2}\right) = \sqrt{\frac{1-\sin x}{1+\sin x}} \] It appears frequently in integration problems involving inverse trigonometric functions.

Answer 1

The Correct Option is D