QJEEM19Ma7cbb: JEE Main 2019 Mathematics - Class 12 JEEM19Ma7cbb Solution: = e secx f(x) + C then a possible choice of f(x) is :-

Question

Verified Solution • 2025–26 Session moderate

\int {{e^{\sec x}}}

(\sec x\tan xf(x) + \sec x\tan x + se{x^2}x)dx

= e^secxf(x) + C then a possible choice of f(x) is :-

Accepted Answer

Given

\int {{e^{\sec x}}\left( {\sec x\tan x\,f(x) + (sec\,x\,tan\,x\, + se{c^2}x)} \right)}

= {e^{\sec x}}f(x) + C

Differentiating both sides with respect to x,

{e^{\sec x}}.\sec x\tan x\,f\left( x \right)

+

{e^{\sec x}}\left( {\sec x\tan x + {{\sec }^2}x} \right)

=

{e^{\sec x}}

. f'(x) + f(x)

{e^{\sec x}}

.

\sec x\tan x

\Rightarrow

{e^{\sec x}}\left( {\sec x\tan x + {{\sec }^2}x} \right)

=

{e^{\sec x}}

. f'(x)

\Rightarrow

f'(x) =

{\sec x\tan x + {{\sec }^2}x}

\therefore

f(x) = \int {\left( {\left( {\sec x\tan x} \right) + {{\sec }^2}x} \right)dx}

\therefore

f(x) = \sec x + \tan x + C

From question we can not find the value of C. So we have to choose any random value of C where (

\sec x + \tan x

) present.

Only in option D, (

\sec x + \tan x

) term present. So only possible option is D.

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