Let f(x)=(2 sin 2 x-1/ cos x)+( cos x(2 sin x+1)/1+ sin x) then ∫ e x ( f ( x )+ f prime( x )) dx where c is the constant of integeration)

Question

Let f(x)=(2 sin 2 x-1/ cos x)+( cos x(2 sin x+1)/1+ sin x) then ∫ e x ( f ( x )+ f prime( x )) dx where c is the constant of integeration) (A)  ex tan x+c (B) e x cot x + c (C) e x operatornamecosec2 x + c (D) e x sec 2 x + c

Tardigrade Admin · Accepted Answer

( cos x(1+2 sin x)/1+ sin x)-( cos 2 x- sin 2 x/ cos x) =( cos 2 x(1+2 sin x)-(1+ sin x)( cos 2 x- sin 2 x)/ cos x(1+ sin x))=( sin x cos 2 x+ sin 3 x+ sin 2 x/ cos x(1+ sin x)) =( sin x( cos 2 x+ sin x)+ sin 2 x/ cos x(1+ sin x))= tan x

Q. Let $f (x) = \frac{2 s i n ^{2} x - 1}{c o s x} + \frac{c o s x ( 2 s i n x + 1 )}{1 + s i n x}$ then $\int e^{x} (f (x) + f^{'} (x)) d x$ where $c$ is the constant of integeration)

$e^{x} tan x + c$

$e^{x} cot x + c$

$e^{x} cosec^{2} x + c$

$e^{x} sec^{2} x + c$

Q. Let f(x)=cosx2sin2x−1​+1+sinxcosx(2sinx+1)​ then ∫ex(f(x)+f′(x))dx where c is the constant of integeration)

extanx+c

excotx+c

excosec2x+c

exsec2x+c

1+sinxcosx(1+2sinx)​−cosxcos2x−sin2x​ =cosx(1+sinx)cos2x(1+2sinx)−(1+sinx)(cos2x−sin2x)​=cosx(1+sinx)sinxcos2x+sin3x+sin2x​ =cosx(1+sinx)sinx(cos2x+sinx)+sin2x​=tanx

Q. Let $f (x) = \frac{2 s i n ^{2} x - 1}{c o s x} + \frac{c o s x ( 2 s i n x + 1 )}{1 + s i n x}$ then $\int e^{x} (f (x) + f^{'} (x)) d x$ where $c$ is the constant of integeration)

$e^{x} tan x + c$

$e^{x} cot x + c$

$e^{x} cosec^{2} x + c$

$e^{x} sec^{2} x + c$