Question

If $I=\underset{0}{\overset{1}{\int }}\frac{dx}{\sqrt{1+x^4}}$, then

• A. $I>1$
• B. $I<\frac{1}{\sqrt{2}}$
• C. $I>\frac{1}{\sqrt{2}}$
• D. $I>2$

Answer 1

The correct option is C $I>\frac{1}{\sqrt{2}}$

As $0<x<1$
$0<x^4<1\Rightarrow 1<\sqrt{1+x^4}<\sqrt{2}\Rightarrow \frac{1}{\sqrt{2}}<\frac{1}{\sqrt{1+x^4}}<1\Rightarrow \underset{0}{\overset{1}{\int }}\frac{1}{\sqrt{2}}dx<I<\underset{0}{\overset{1}{\int }}1dx\Rightarrow \frac{1}{\sqrt{2}}<I<1$