Solution To Problem 0002 | Higher Secondary Mathematics

Problem 0002

Prove that $I = \int\limits_0^{\pi /2} {{{\sqrt {\sec x} } \over {\sqrt {\cos ecx  }+ \sqrt {\sec x} }}} dx = {\pi  \over 4}$

Answer:

$I = \int\limits_0^{\pi /2} {{{\sqrt {\sec x} } \over {\sqrt {\cos ec x }+ \sqrt {\sec x} }}} dx$....... (1)

or $I = \int\limits_0^{\pi /2} {{{\sqrt {\sec (\pi /2 - x)} } \over {\sqrt {\cos ec(\pi /2 - x)}  + \sqrt {\sec (\pi /2 - x} )}}} dx$

or $I = \int\limits_0^{\pi /2} {{{\sqrt {\cos ecx} } \over {\sqrt {\sec x + } \sqrt {\cos ecx} }}} dx$ ....... (2)

By (1) + (2) we get

$2I = \int\limits_0^{\pi /2} {dx = {\pi  \over 2}}$

There fore $I = {\pi  \over 4}$       ....(Proved)