Question

Evaluate the following integral: $ \int e^{2\theta} \left( 2 \cos^2 \theta - \sin 2\theta \right) \, d\theta $

• A. \( \frac{e^{2\theta}}{2} \left( 2 \cos^2 \theta - \sin 2\theta \right) \)
• B. \( e^{2\theta} \cos^2 \theta - \frac{1}{2} \sin 2\theta \)
• C. \( e^{2\theta} ( \cos^2 \theta - \sin^2 \theta ) \)
• D. \( \frac{e^{2\theta}}{2} ( \cos^2 \theta + \sin^2 \theta ) \)

Hint:

For trigonometric integrals, use identities like \( \sin 2\theta = 2 \sin \theta \cos \theta \) to simplify the integrand before performing the integration.

Answer 1

The Correct Option is B

We are given the integral: \[ \int e^{2\theta} \left( 2 \cos^2 \theta - \sin 2\theta \right) \, d\theta \] First, note that: \[ \sin 2\theta = 2 \sin \theta \cos \theta \] So, the integral becomes: \[ \int e^{2\theta} \left( 2 \cos^2 \theta - 2 \sin \theta \cos \theta \right) \, d\theta \] We can now split this into two integrals: \[ 2 \int e^{2\theta} \cos^2 \theta \, d\theta - 2 \int e^{2\theta} \sin \theta \cos \theta \, d\theta \] The second term is straightforward: \[ \int e^{2\theta} \sin \theta \cos \theta \, d\theta = \frac{1}{2} e^{2\theta} \sin 2\theta \]
Thus, the result of the integration is: \[ e^{2\theta} \cos^2 \theta - \frac{1}{2} \sin 2\theta \]
Thus, the correct answer is \( e^{2\theta \cos^2 \theta - \frac{1}{2} \sin 2\theta} \).