STD 12 - 7.2 definite integral — JEE कक्षा 11 साइन्स — Question

$ = \int_0^{\pi /2} {\frac{{dx}}{{2{{\sin }^2}\frac{x}{2} + 2{{\cos }^2}\frac{x}{2} + {{\cos }^2}\frac{x}{2} - {{\sin }^2}\frac{x}{2}}}} $

$ = \int_0^{\pi /2} {\frac{{dx}}{{{{\sin }^2}\frac{x}{2} + 3{{\cos }^2}\frac{x}{2}}}} $

$= \int_0^{\pi /2} {\frac{{{{\sec }^2}\frac{x}{2}}}{{3 + {{\tan }^2}\frac{x}{2}}}dx} $

$t = \tan \frac{x}{2}$ रखने पर,

$ \Rightarrow dt = \frac{1}{2}{\sec ^2}\frac{x}{2}dx$

$I = 2\int_0^1 {\frac{{dt}}{{3 + {t^2}}} = \frac{2}{{\sqrt 3 }}{{\tan }^{ - 1}}\left( {\frac{1}{{\sqrt 3 }}} \right)} $.