Question

If \( \int_0^{2\pi} (\sin^4 x + \cos^4 x) \, dx = K \int_0^\pi \sin^2 x \, dx + L \int_0^\frac{\pi}{2} \cos^2 x \, dx \) and \( K, L \in \mathbb{N} \), then the number of possible ordered pairs \( (K, L) \) is}

• A. \( 1 \)
• B. \( 2 \)
• C. \( 3 \)
• D. \( 4 \)

Hint:

When evaluating integrals involving powers of sine and cosine, use the power-reduction formulas to simplify the integrand.

Answer 1

The Correct Option is B

Step 1: Evaluating \( \int_0^{2\pi} (\sin^4 x + \cos^4 x) \, dx \) 

\[ \sin^4 x + \cos^4 x = (\sin^2 x + \cos^2 x)^2 - 2 \sin^2 x \cos^2 x \] \[ = 1 - 2 \sin^2 x \cos^2 x \] \[ = 1 - \frac{1}{2} (2 \sin x \cos x)^2 \] \[ = 1 - \frac{1}{2} \sin^2 (2x) \] \[ = 1 - \frac{1}{2} \cdot \frac{1 - \cos (4x)}{2} \] \[ = 1 - \frac{1}{4} (1 - \cos (4x)) \] \[ = \frac{3}{4} + \frac{1}{4} \cos (4x) \]

Step 2: Integrating the Expression

\[ \int_0^{2\pi} (\sin^4 x + \cos^4 x) \, dx = \int_0^{2\pi} \left( \frac{3}{4} + \frac{1}{4} \cos (4x) \right) \, dx \] \[ = \left[ \frac{3}{4} x + \frac{1}{16} \sin (4x) \right]_0^{2\pi} \] \[ = \frac{3}{4}(2\pi) + \frac{1}{16} \sin (8\pi) - \left( \frac{3}{4}(0) + \frac{1}{16} \sin (0) \right) \] \[ = \frac{3\pi}{2} \]

Step 3: Evaluating \( \int_0^\pi \sin^2 x \, dx \)

\[ \int_0^\pi \sin^2 x \, dx = \int_0^\pi \frac{1 - \cos (2x)}{2} \, dx \] \[ = \left[ \frac{x}{2} - \frac{\sin (2x)}{4} \right]_0^\pi \] \[ = \frac{\pi}{2} - \frac{\sin (2\pi)}{4} - \left( \frac{0}{2} - \frac{\sin (0)}{4} \right) \] \[ = \frac{\pi}{2} \]

Step 4: Evaluating \( \int_0^{\pi/2} \cos^2 x \, dx \)

\[ \int_0^{\pi/2} \cos^2 x \, dx = \int_0^{\pi/2} \frac{1 + \cos (2x)}{2} \, dx \] \[ = \left[ \frac{x}{2} + \frac{\sin (2x)}{4} \right]_0^{\pi/2} \] \[ = \frac{\pi}{4} + \frac{\sin \pi}{4} - \left( \frac{0}{2} + \frac{\sin 0}{4} \right) \] \[ = \frac{\pi}{4} \]

Step 5: Comparing and Finding \( K \) and \( L \)

\[ \frac{3\pi}{2} = K \cdot \frac{\pi}{2} + L \cdot \frac{\pi}{4} \] Dividing by \( \frac{\pi}{4} \): \[ 6 = 2K + L \] Possible solutions:

• \( K = 1 \), \( L = 4 \)
• \( K = 2 \), \( L = 2 \)

Thus, there are 2 possible ordered pairs \( (K, L) \).