-2- x x-1-2 x2 d xA- 2 cosec x- x-CB- 1-2 cosec1- x-CC- 1-2 cosec x- x-CD- 2 cosec x- x-C

The correct option is B 1/2cosec x+cot x+C∫(2cos x+1)/(2+cos x)^2dx=∫(2+cos x)cos x+sin^2x/(2+cos x)^2dx =∫cos x/2+cos xdx ∫ sin^2x/(2+cos x)^2dx=sin x/2+cos ...

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Byju's Answer

Standard XII

Mathematics

Special Integrals -4

∫2+ x x/1+2 x...

Question

$\int \frac{(2 + s e c x) s e c x}{(1 + 2 s e c x)^{2}} d x$

\frac{1}{2 c o s e c x + c o t x} + C

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2 c o s e c x + c o t x + C

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\frac{1}{2 c o s e c x - c o t x} + C

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2 c o s e c x - c o t x + C

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Solution

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

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∫(2+sec x)sec x(1+2sec x)2dx

The correct option is A 12cosec x+cot x+C∫(2cos x+1)(2+cos x)2dx=∫(2+cos x)cos x+sin2x(2+cos x)2dx=∫cos x2+cos xdx−∫−sin2x(2+cos x)2dx=sin x2+cos x+c

$\int \frac{(2 + s e c x) s e c x}{(1 + 2 s e c x)^{2}} d x$