Question

The value of \( \int_0^{2\pi} (\sin^4 x + \cos^4 x) \, dx \) is equal to:

• A. \( 15\pi \)
• B. \( \frac{15\pi}{2} \)
• C. \( 25\pi \)
• D. \( \frac{3\pi}{2} \)

Hint:

For integrals of \( \sin^{2n}x + \cos^{2n}x \) over \( [0, 2\pi] \), always use the identity \( (a+b)^2 - 2ab \) to reduce the power of the trigonometric functions.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We simplify the integrand using trigonometric identities. The expression \( \sin^4 x + \cos^4 x \) can be converted into a form involving \( \cos 4x \).
Step 2: Key Formula or Approach:
\[ \sin^4 x + \cos^4 x = (\sin^2 x + \cos^2 x)^2 - 2\sin^2 x \cos^2 x \] \[ = 1 - \frac{1}{2}(4\sin^2 x \cos^2 x) = 1 - \frac{1}{2}\sin^2 2x \] \[ = 1 - \frac{1}{2} \left( \frac{1 - \cos 4x}{2} \right) = \frac{3}{4} + \frac{1}{4}\cos 4x \]
Step 3: Detailed Explanation:
The integral becomes: \[ \int_0^{2\pi} \left( \frac{3}{4} + \frac{1}{4}\cos 4x \right) dx \] \[ = \left[ \frac{3}{4}x + \frac{1}{16}\sin 4x \right]_0^{2\pi} \] \[ = \left( \frac{3}{4}(2\pi) + 0 \right) - (0 + 0) \] \[ = \frac{3\pi}{2} \]
Step 4: Final Answer:
The value is \( \frac{3\pi}{2} \).