Question

Evaluate the integral: \[ \int \frac{\cos \theta}{2 - \sin^2 \theta} \, d\theta \]

• A. \( \frac{\cos \theta}{2} \)
• B. \( \frac{\sin \theta}{2} \)
• C. \( \frac{\sin \theta}{4} \)
• D. \( \frac{1}{2} \ln |\cos \theta| \)

Hint:

For integrals involving trigonometric functions, simplify the expression by using trigonometric identities to reduce it to a more manageable form.

Answer 1

The Correct Option is D

We start by simplifying the denominator using the trigonometric identity: \[ 2 - \sin^2 \theta = \cos^2 \theta + 1 \] So the integral becomes: \[ \int \frac{\cos \theta}{\cos^2 \theta + 1} \, d\theta \] This simplifies further to: \[ \frac{1}{2} \int \frac{d(\cos \theta)}{\cos^2 \theta + 1} \] Using the standard formula for the integral of the secant function, we get: \[ \frac{1}{2} \ln |\cos \theta| \]