Question

Evaluate the integral: \[ I = \int_0^{3\pi/2} \frac{\cos^5 x}{\cos^3 x+\sin^3 x}dx \]

• A. \( 0 \)
• B. \( 1 \)
• C. \( \frac{\pi}{4} \)
• D. \( \frac{3\pi}{4} \)

Hint:

For definite integrals involving trigonometric functions, check for symmetry in the interval to simplify evaluation.

Answer 1

The Correct Option is A

Using symmetry properties of trigonometric functions: \[ I = \int_0^{3\pi/2} \frac{\cos^5 x}{\cos^3 x+\sin^3 x}dx \] Splitting the integral into symmetric regions and simplifying, \[ I = 0 \] Thus, the correct answer is: \[ 0 \]