Question

For $f(x)=x^4+|x|$, let $I_1=\underset{0}{\overset{\pi }{\int }}f(\cos x)dx$ and $I_2=\underset{0}{\overset{\pi /2}{\int }}f(\sin x)dx$ then $\frac{I_1}{I_2}$ has the value equal to

• A. 1
• B. 1 / 2
• C. 2
• D. 4

Answer 1

Answer: (C)

Clearly f is an even function, hence
$I_1=\underset{0}{\overset{\pi }{\int }}f(\cos (\pi -x)dx=\underset{0}{\overset{\pi }{\int }}f(-\cos x)dx={\int }_0^{\pi }f(\cos x)dx$
$\therefore I_1=2\underset{0}{\overset{\pi /2}{\int }}f(\cos x)dx=2\underset{0}{\overset{\pi /2}{\int }}f(\sin x)dx=2I_2\Rightarrow \frac{I_1}{I_2}=2$