Question

$\underset{-\frac{\pi }{2}}{\overset{\frac{\pi }{2}}{\int }}\frac{{\cos }^{2}2x}{1+25^x}dx=$ _______

• A. $\frac{\pi }{4}$
• B. $-\frac{\pi }{2}$
• C. $\frac{\pi }{2}$
• D. $-\frac{\pi }{4}$

Answer 1

Answer: (A)

$\underset{-\frac{\pi }{2}}{\overset{\frac{\pi }{2}}{\int }}\frac{{\cos }^{2}2x}{1+25^x}dx=\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{{\cos }^{2}2x}{1+25^x}+\frac{{\cos }^{2}2x}{1+25^{-x}}dx$
$=\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\cos }^{2}2xdx$
$=\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{1+\cos 4x}{2}dx$
$=\frac{1}{2}{\left[x+\frac{\sin 4x}{4}\right]}_0^{\frac{\pi }{2}}=\frac{\pi }{4}$