- e x 1-sin x -1-cos x dx is-A- extanx-2-CB- tanx-2-CC- ex-CD- e x sin x - C

The correct option is A e^xtan ( x/2 )+C ex (1+sinx/1+cosx nbsp; )=1/2e^x (1+tanx/2 nbsp; )^2 =1/2e^x (sec^2x/2+2 tan x/2 nbsp; )=e^x(f^1(x)+f(x)) ...

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Byju's Answer

Standard XII

Mathematics

Differentiation of Inverse Trigonometric Functions

∫ e x 1+sin x...

Question

$\int e^{x} (\frac{1 + s i n x}{1 + c o s x})$ dx is......................

$e^{x} t a n (\frac{x}{2}) + C$

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$t a n (\frac{x}{2}) + C$

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$e^{x} + C$

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$e^{x} s i n x + C$

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Solution

The correct option is A
$e^{x} t a n (\frac{x}{2}) + C$

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

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Similar questions

\int e^{x} (\frac{1 + sin x}{1 + cos x}) d x

$\int e^{x}$ (sin x + cos x)dx is equal to -

\int e^{x} (\frac{1 - \sin x}{1 - \cos x}) d x

(a)

- e^{x} \tan \frac{x}{2} + C

(c)

- \frac{1}{2} e^{x} \tan \frac{x}{2} + C

(d)

- \frac{1}{2} e^{x} \cot \frac{x}{2} + C

Q. The solution of the differential equation

[(cos x) d x - d y] (1 + x^{2}) + e^{x} [(1 + x^{2}) {tan}^{- 1} x + 1] d x = 0

is
(where

C

is arbitrary constant)

\int \frac{e^{x} (1 + x)}{\cos^{2} (x e^{x})} d x =

(a) 2 log_e cos (xe^x) + C

(b) sec (xe^x) + C

(c) tan (xe^x) + C

(d) tan (x + e^x) + C

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∫ex(1+sin x1+cos x) dx is......................

The correct option is A extan(x2)+C ex(1+sinx1+cosx)=12ex(1+tanx2)2=12ex(sec2x2+2tanx2)=ex(f1(x)+f(x))

$\int e^{x} (\frac{1 + s i n x}{1 + c o s x})$ dx is......................

The correct option is A
$e^{x} t a n (\frac{x}{2}) + C$

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