Question

If \( I_1 = \int \sin^6 x \, dx \) and \( I_2 = \int \cos^6 x \, dx \) then \( I_1 + I_2 = \)

• A. \( \frac{5x}{8} + \frac{3\cos 4x}{32} + c \)
• B. \( \frac{1}{32} (20x - 3\sin 4x) + c \)
• C. \( \frac{1}{32} (20x + 3\sin 4x) + c \)
• D. \( \frac{5x}{4} + \frac{3\sin 4x}{16} + c \)

Hint:

The identities for sums of powers of sine and cosine are very useful. Remember \( \sin^4x+\cos^4x = 1 - 2\sin^2x\cos^2x \) and \( \sin^6x+\cos^6x = 1 - 3\sin^2x\cos^2x \). These allow quick simplification before integration.

Answer 1

The Correct Option is C