Question

${\displaystyle {\int }_0^{\frac{\pi }{2}}\left(\frac{3\sqrt{\sec x}}{3\sqrt{\sec x}+3\sqrt{\cos ecx}}\right)dx}$ = ______ 

Options

• ${\displaystyle \frac{\pi }{2}}$

• ${\displaystyle \frac{\pi }{3}}$

• ${\displaystyle \frac{\pi }{4}}$

• ${\displaystyle \frac{\pi }{6}}$

Answer 1

${\displaystyle {\int }_0^{\frac{\pi }{2}}\left(\frac{3\sqrt{\sec x}}{3\sqrt{\sec x}+3\sqrt{\cos ecx}}\right)dx}$ = ${\displaystyle \underline{\frac{\pi }{4}}}$.

Explanation:

 Let I = ${\displaystyle {\int }_0^{\frac{\pi }{2}}\left(\frac{3\sqrt{\sec x}}{3\sqrt{\sec x}+3\sqrt{\cos ecx}}\right)dx}$ ............(i)

∴ I = ${\displaystyle {\int }_0^{\frac{\pi }{2}}\left(\frac{3\sqrt{\cos ecx}}{3\sqrt{\cos ecx}+3\sqrt{\sec x}}\right)}$dx ................(ii) ............${\displaystyle [\because {\int }_0^{a}f\left(x\right)dx={\int }_0^a(a-x)dx]}$

Adding (i) and (ii), we get

2I = ${\displaystyle {\int }_0^{\frac{\pi }{2}}dx={\left[x\right]}_0^{\frac{\pi }{2}}=\frac{\pi }{2}\Rightarrow I=\frac{\pi }{4}}$