Question

Evaluate the integral: \(\int \frac{\sin x}{\sin 4x} \, dx\)

• A. \( -\frac{1}{8}\log \left| \frac{1+\sin x}{1-\sin x} \right| + \frac{1}{4\sqrt{2}}\log \left| \frac{1+\sqrt{2}\sin x}{1-\sqrt{2}\sin x} \right| + C \)
• B. \( \frac{1}{8}\log \left| \frac{1+\sin x}{1-\sin x} \right| - \frac{1}{4\sqrt{2}}\log \left| \frac{1+\sqrt{2}\sin x}{1-\sqrt{2}\sin x} \right| + C \)
• C. \( -\frac{1}{8}\log \left| \frac{1+\sin x}{1-\sin x} \right| + \frac{1}{4\sqrt{2}}\log \left| \frac{1-\sqrt{2}\sin x}{1+\sqrt{2}\sin x} \right| + C \)
• D. None of these

Hint:

When you see \( \sin nx \) in the denominator and \( \sin x \) in the numerator, expanding \( \sin nx \) often allows the \( \sin x \) term to cancel, leading to an integral that can be solved via the \( \sin x = t \) substitution.

Answer 1

The Correct Option is A