Question

Let $J=\underset{0}{\overset{1}{\int }}\frac{x}{1+x^8}dx.$
Consider the following assertions:
I. $J>\frac{1}{4}$

II. $J<\frac{\pi }{8}$
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

For $0<x<1$
$x^4>x^8\Rightarrow 1+x^4>1+x^8\Rightarrow \frac{1}{1+x^4}<\frac{1}{1+x^8}\Rightarrow \frac{x}{1+x^4}<\frac{x}{1+x^8}\Rightarrow \underset{0}{\overset{1}{\int }}\frac{x}{1+x^4}dx<\underset{0}{\overset{1}{\int }}\frac{x}{1+x^8}dx\Rightarrow J>\underset{0}{\overset{1}{\int }}\frac{x}{1+x^4}dx\Rightarrow J>\frac{\pi }{8}>\frac{1}{4}$