QJEEM17Ma04d1: JEE Main 2017 Mathematics - Class 12 JEEM17Ma04d1 Solution: The integral is equal to

Question

Verified Solution • 2025–26 Session moderate

The integral

\int\limits_{{\pi \over 4}}^{{{3\pi } \over 4}} {{{dx} \over {1 + \cos x}}}

is equal to

Accepted Answer

\int\limits_{{\pi \over 4}}^{{{3\pi } \over 4}} {{{dx} \over {1 + \cos x}}}

=

\int\limits_{{\pi \over 4}}^{{{3\pi } \over 4}} {{{dx} \over {2{{\cos }^2}{x \over 2}}}}

=

{1 \over 2}\int\limits_{{\pi \over 4}}^{{{3\pi } \over 4}} {{{\sec }^2}} {x \over 2}\,dx

{1 \over 2}\left[ {{{\tan {x \over 2}} \over {{1 \over 2}}}} \right]_{{\pi \over 4}}^{{{3\pi } \over 4}}

=

\left[ {\tan {x \over 2}} \right]_{{\pi \over 4}}^{{3 \over 4}}

= tan

{{3\pi } \over 8}

-

tan

{\pi \over 8}

=

\left( {\sqrt 2 + 1} \right) - \left( {\sqrt 2 - 1} \right)

= 2

Step-by-Step Verified Solution