_ ^ e^x (1 + x) x ;dx = - Vidyadip

MCQ

$\int_{}^{} {{e^x}(1 + \tan x)\sec x\;dx = } $

A
${e^x}\cot x$
B
${e^x}\tan x$
✓
${e^x}\sec x$
D
${e^x}\cos x$

✓

Answer

Correct option: C.

${e^x}\sec x$

c
(c)$\int_{}^{} {{e^x}(1 + \tan x)\sec x\,dx} = \int_{}^{} {{e^x}\sec x\,dx} + \int_{}^{} {{e^x}\tan x\sec x\,dx} $
$ = {e^x}\sec x - \int_{}^{} {{e^x}\sec x\tan x\,dx} + \int_{}^{} {{e^x}\sec x\tan x\,dx} $
$ = {e^x}\sec x + c.$
Aliter : $\int_{}^{} {{e^x}(\sec x + \sec x\tan x)\,dx} = {e^x}\sec x + c$
Obviously, it is of the form $\int_{}^{} {{e^x}\left\{ {f(x) + f'(x)} \right\}} \,dx.$

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