Question

The value of $\int_{a}^{a + (\pi/2)}{(\sin^{4}x + \cos^{4}x)dx}$ is

• A. Independent of $a$
• B. $a\left( \frac{\pi}{2} \right)^{2}$
• C. $\frac{3\pi}{8}$
• D. $\frac{3\pi a^{2}}{8}$

Answer 1

Answer: (C)

$\frac{3\pi}{8}$

Answer 2

Since $\sin^{4}x + \cos^{4}x$is a periodic function with period $\frac{\pi}{2},$ therefore $\int_{a}^{a + (\pi/2)}{(\sin^{4}x + \cos^{4}x)\ dx}$

$= \int_{0}^{\pi/2}{(\sin^{4}x + \cos^{4}x)dx} = 2\int_{0}^{\pi/2}{\sin^{4}xdx = \frac{3\Gamma(5/2)\Gamma(1/2)}{2\Gamma\left( \frac{4 + 0 + 2}{2} \right)} = \frac{3\pi}{8}}$.