Question

If $J=\underset{0}{\overset{1}{\int }}\frac{x}{1+x^8}dx$
Consider the following statements
$I)J>\frac{1}{4}II)J>\frac{\pi }{4}$
Then

• A. Only $I$ is true
• B. Only $II$ is true
• C. Both $I$ and $II$ are true
• D. Neither $I$ nor $II$ is true

Answer 1

The correct option is A Only $I$ is true

$J=\underset{0}{\overset{1}{\int }}\frac{x}{1+x^8}dx\because 0<x^8<1$
For $x\in (0,1)$, we have
$1>x^8>0\Rightarrow 2>1+x^8>1\Rightarrow \frac{1}{2}<\frac{1}{1+x^8}<1\Rightarrow \frac{x}{2}<\frac{x}{1+x^8}<x$
Now,
$\underset{0}{\overset{1}{\int }}\frac{x}{2}dx<\underset{0}{\overset{1}{\int }}\frac{x}{1+x^8}dx<\underset{0}{\overset{1}{\int }}xdx\Rightarrow \frac{1}{4}<J<\frac{1}{2}<\frac{\pi }{4}$