Question

Let $f(x)=x^4+|x|$ and $I_1=\underset{0}{\overset{\pi }{\int }}f(\cos x)dx$ and $I_2=\underset{0}{\overset{\frac{\pi }{2}}{\int }}f(\sin x)dx$, then the value of $\frac{I_1}{I_2}$ is

• A. 1
• B. $\frac{1}{2}$
• C. 2
• D. $\frac{1}{4}$

Answer 1

Answer: (C)

$I_1=\underset{0}{\overset{\pi }{\int }}\left({\cos }^{4}x+|\cos x|\right)dx$
$I_1=2\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\cos }^{4}x+|\cos x|dx$
$\downarrow \text{ king }$
$I_1=2\underset{0}{\overset{\frac{\pi }{2}}{\int }}\left({\sin }^{4}x+|\sin x|\right)dx=2I_2\Rightarrow \frac{I_1}{I_2}=2$