∫ ( 2 + s e c x ) s e c x ( 1 + 2 s e c x ) 2 d x Options: A . &nbsp 1 2 c o s e c x + c o t x + C B . &nbsp 2 c o s e c x + c o t x + C C . &nbsp 1 2 c o s e c x − c o t x + C D . &nbsp 2 c o s e c x − c o t x + C

Answer: Option A : A ∫ ( 2 c o s x + 1 ) ( 2 + c o s x ) 2 d x = ∫ ( 2 + c o s x ) c o s x + s i n 2 x ( 2 + c o s x ) 2 d x = ∫ c o s x 2 + c o s x d x − ∫ − s i n 2 x ( 2 + c o s x ) 2 d x = s i n x 2 + c o s x + c

∫(2+sec x)sec x(1+2sec x)2dx

Discussion Forum : Indefinite Integration

Question -

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

Options:

A .

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

B .

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

C .

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

D .

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

Answer: Option A
:
A

\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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