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int("cosec x")/(cos^(2)(1+log tan.(x)/(2...

`int("cosec x")/(cos^(2)(1+log tan.(x)/(2)))dx` is equal to

A

`sin^(2)[1+log tan.(x)/(2)]+C`

B

`tan[1+log tan.(x)/(2)]+C`

C

`sec^(2)[1+log tan.(x)/(2)]+C`

D

`-tan[1+log tan.(x)/(2)]+C`

Text Solution

Verified by Experts

The correct Answer is:
B
Let `l=int("cosec x")/(cos^(2)(1+log tan.(x)/(2)))dx`
Put `1+log tan.(x)/(2)=t`
`rArr (1)/(tan.(x)/(2)).sec^(2).(x)/(2).(1)/(2)dx=dt rArr" cosec x dx = dt "` ltBrgt `therefore" "l=int(dt)/(cos^(2)t)=int sec^(2)t dx=tant+C`
`=tan(1+log tan.(x)/(2))+C`
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