QJEEM16Ma7e73: JEE Main 2016 Mathematics - Class 12 JEEM16Ma7e73 Solution: If    where k is a constant of integration, then A +…

Question

Verified Solution • 2025–26 Session moderate

If

\int {{{dx} \over {{{\cos }^3}x\sqrt {2\sin 2x} }}} = {\left( {\tan x} \right)^A} + C{\left( {\tan x} \right)^B} + k,

where k is a constant of integration, then A + B +C equals :

Accepted Answer

\int {{{dx} \over {{{\cos }^3}x\sqrt {2\sin 2x} }}}

=

\int {{{dx} \over {{{\cos }^3}x\sqrt {4\sin x\cos x} }}}

=

\int {{{dx} \over {2{{\cos }^4}x\sqrt {\tan x} }}}

Let tan x = t²

\Rightarrow

\,\,\,

sec²xdx = 2t dt

as sec²x = 1 + tan²x = 1 + t⁴

=

\int {{{{{\sec }^4}x\,dx} \over {2\sqrt {\tan x} }}}

=

\int {{{{{\sec }^2}x\left( {{{\sec }^2}x\,dx} \right)} \over {2\sqrt {\tan x} }}}

=

\int {{{\left( {1 + {t^4}} \right)2t\,dt} \over {2t}}}

=

\int {\left( {1 + {t^4}} \right)} \,dt

= t +

{{{t^5}} \over 5}

+ k

=

\sqrt {\tan x}

+

{1 \over 5}

tan

^{{5 \over 2}}

x + k

By comparing with the given equation, we get

A =

{1 \over 2}

, B =

{5 \over 2}

, C =

{1 \over 5}

\therefore\,\,\,

A + B + C =

{{16} \over 5}

QJEEM16Ma7e73: JEE Main 2016 Mathematics - Class 12 JEEM16Ma7e73 Solution: If &nbsp;&nbsp; where k is a constant of integration, then A +…