If \int_{0}^{\pi/2}(\sin^{4}x+\cos^{4}x)\,dx=k then what is the value of \int_{0}^{20\pi}(\sin^{4}x+\cos^{4}x)\,dx ?
- A. k
- B. 10k
- C. 20k
- D. 40k ✓
Correct Answer: D. 40k
Explanation
The function f(x) = \sin^4 x + \cos^4 x is periodic with a period of \frac{\pi}{2}. The interval from 0 to 20\pi contains exactly 40 periods of length \frac{\pi}{2}. By the property of periodic functions, \int_{0}^{nT} f(x)\,dx = n \int_{0}^{T} f(x)\,dx, so the integral is 40k.
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