Question

$\int e^{2\theta}(2\cos^{2}\theta-\sin 2\theta)\, d\theta =$

• A. $e^{2\theta}\cos^{2}\theta+C$
• B. $e^{2\theta}\sin 2\theta+C$
• C. $2e^{2\theta}\cos^{2}\theta+C$
• D. $e^{2\theta}\sin \theta+C$
• E. $e^{2\theta}\cos 2\theta+C$

Hint:

$\int e^x (f(x) + f'(x)) dx = e^x f(x) + C$ is a very common identity in entrance exams.

Answer 1

The Correct Option is A

Step 1: Concept
Recognize the form $\int e^{kx} [kf(x) + f'(x)] dx = e^{kx}f(x) + C$.

Step 2: Analysis
Let $f(\theta) = \cos^2\theta$ and $k=2$. $f'(\theta) = 2\cos\theta(-\sin\theta) = -\sin 2\theta$.

Step 3: Calculation The integral matches $e^{2\theta}[2f(\theta) + f'(\theta)]$. Result $= e^{2\theta}\cos^2\theta + C$. Final Answer: (A)