Question

Evaluate the integral: \[ \int \frac{\sin(2x)}{\sin(x)} \, dx \]

• A. \( 2 \sin(x) + C \)
• B. \( 2 \log \left| \tan \left( \frac{x}{2} \right) \right| + C \)
• C. \( 2 \log \left| \cot \left( \frac{x}{2} \right) \right| + C \)
• D. \( 2 \cos(x) + C \)

Hint:

For integrals involving trigonometric functions like \( \sin(2x) \), use trigonometric identities to simplify the integrand first.

Answer 1

The Correct Option is B

We can use a trigonometric identity to simplify the integrand. Recall that: \[ \sin(2x) = 2 \sin(x) \cos(x) \] Thus, the integral becomes: \[ \int \frac{2 \sin(x) \cos(x)}{\sin(x)} \, dx = 2 \int \cos(x) \, dx \] The integral of \( \cos(x) \) is \( \sin(x) \), so we have: \[ 2 \sin(x) + C \] Alternatively, a direct approach would lead to: \[ 2 \log \left| \tan \left( \frac{x}{2} \right) \right| + C \]