Let f(x)=2sin2x−1cosx+cosx(2sinx+1)1+sinx then ∫ex(f(x)+f′(

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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

equals
(where c is the constant of integeration)

Options:

A . extanx+c

B . excotx+c

C . excosec2x+c

D . None of these

Answer: Option A
:
A

\frac{c o s x (1 + 2 s i n x)}{1 + s i n x} - \frac{c o s^{2} x - s i n^{2} x}{c o s x} = \frac{c o s^{2} x (1 + 2 s i n x) - (1 + s i n x) (c o s^{2} x - s i n^{2} x)}{c o s x (1 + s i n x)}

= \frac{s i n x c o s^{2} x + s i n^{2} x (1 + s i n x)}{c o s x (1 + s i n x)} = \frac{s i n x (1 - s i n x) + s i n^{2} x}{c o s x} = t a n x

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