What is I equal to?
Consider the following for the next two (02) items that follow: Let I=\int_{-2\pi}^{2\pi}\frac{\sin^{4}x+\cos^{4}x}{1+3^{x}}dx
- A. 0
- B. \frac{3\pi}{4}
- C. \frac{3\pi}{2} ✓
- D. 3\pi
Correct Answer: C. \frac{3\pi}{2}
Explanation
Use the property \int_{-a}^a \frac{f(x)}{1+k^x} dx = \int_0^a f(x) dx for even functions f(x). Since \sin^4 x + \cos^4 x is an even function, I = \int_0^{2\pi} (\sin^4 x + \cos^4 x) dx. The function has a period of \frac{\pi}{2}, so the integral evaluates to 4 \int_0^{\pi/2} (\sin^4 x + \cos^4 x) dx = 2 \int_0^\pi (\sin^4 x + \cos^4 x) dx. From the previous question, \int_0^\pi (\sin^4 x + \cos^4 x) dx = \frac{3\pi}{4}. Thus, I = 2 \times \frac{3\pi}{4} = \frac{3\pi}{2}.
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