Question

Evaluate the integral \[ \int \sqrt{\csc x - 1} \, dx = \]

• A. \(\log \sin x + \sqrt{\sin^2 x + \sin x} + c\)
• B. \(\log \sin x + 1 + 2\sqrt{\sin^2 x + \sin x} + c\)
• C. \(\log \sin x + \frac{1}{2} + \sqrt{\sin^2 x + \sin x} + c\)
• D. \(\log \sin x + \frac{1}{2} + \sqrt{\sin^2 x + \frac{1}{2}} + \sin x + c\)

Hint:

When solving integrals involving square roots of trigonometric functions, remember to use appropriate trigonometric identities and substitution to simplify the expression before proceeding.

Answer 1

The Correct Option is C

To solve the integral, we first simplify the integran(D) Using substitution techniques and trigonometric identities, we reach the conclusion that the solution is given by Option (3).