Question

If $I={\int }_0^{\frac{\pi }{2}}\frac{dx}{\sqrt{1+si{n}^{3}x}},$ then

• A. 0<I<1
• B. I>$\frac{\pi }{2\sqrt{2}}$
• C. I<$\sqrt{2}$$\pi$
• D. I>2$\pi$

Answer 1

The correct option is B

I>$\frac{\pi }{2\sqrt{2}}$

Since, $xϵ[0,\frac{\pi }{2}]\Rightarrow 1\le 1+si{n}^{3}x\le 2\Rightarrow \frac{1}{\sqrt{2}}\le \frac{1}{\sqrt{1+si{n}^{3}x}}\le 1\Rightarrow {\int }_0^{\frac{\pi }{2}}\frac{1}{\sqrt{2}}dx\le {\int }_0^{\frac{\pi }{2}}\frac{dx}{\sqrt{1+si{n}^{3}x}}\le {\int }_0^{\frac{\pi }{2}}dx\Rightarrow \frac{\pi }{2\sqrt{2}}\le I\le \frac{\pi }{2}.$