Let fx-2sin2 x-1-cos x- cosx2sin x-1-1-sin x then - exfx-f-xdx equalswhere c is the constant of integeration

Cos x(1+2 sin x)/1+sin x cos^2x sin^2x/cos x =cos^2 x(1+2 sin x) (1+sinx)(cos^2 x sin^2 x)/cos x(1+sin x) =sin x cos^2x+sin^2x(1+sin x)/cos x (1+sin x)=sin x( ...

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Byju's Answer

Standard XII

Mathematics

Special Integrals - 1

Let fx=2sin2 ...

Question

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)

e^{x} t a n x + c

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e^{x} c o t x + c

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e^{x} c o s e c^{2} x + c

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None of these

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Solution

The correct option is A $e^{x} t a n x + c$
$\frac{c o s x (1 + 2 s i n x)}{1 + s i n x} - \frac{c o s^{2} x - s i n^{2} x}{c o s x} = \frac{c o s^{2} x (1 + 2 s i n x) - (1 + s i n x) (c o s^{2} x - s i n^{2} x)}{c o s x (1 + s i n x)}$
$= \frac{s i n x c o s^{2} x + s i n^{2} x (1 + s i n x)}{c o s x (1 + s i n x)} = \frac{s i n x (1 - s i n x) + s i n^{2} x}{c o s x} = t a n x$

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Similar questions

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

equals
(where c is the constant of integeration)

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

equals
(where c is the constant of integeration)

Q. Let

f (x) = \frac{2 {sin}^{2} x - 1}{cos x} + \frac{cos x (2 sin x + 1)}{(1 + sin x)}

then

\int e^{x} (f (x) + f^{'} (x)) d x

where

c

is the constant of integration

Q. Let

f (x) = \frac{2 {sin}^{2} x - 1}{cos x} + \frac{cos x (2 sin x + 1)}{1 + sin x} .

Then

\int e^{x} (f (x) + f^{'} (x)) d x

equals
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Q. Assertion :STATEMENT-1 :

\int e^{x} sin x d x = \frac{e^{x}}{2} (sin x - cos x) + c .

Reason: STATEMENT-2 :

\int e^{x} (f (x) + f^{'} (x)) d x = e^{x} f (x) + c .

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Let f(x)=2sin2x−1cosx+cosx(2sinx+1)1+sinx then ∫ex(f(x)+f′(x))dx equals (where c is the constant of integeration)

The correct option is A extanx+ccosx(1+2sinx)1+sinx−cos2x−sin2xcosx=cos2x(1+2sinx)−(1+sinx)(cos2x−sin2x)cosx(1+sinx) =sinxcos2x+sin2x(1+sinx)cosx(1+sinx)=sinx(1−sinx)+sin2xcosx=tanx

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)