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

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

A

`cotx +cot^(-1)x+C`

B

`-e^(ln tan^(-1))x-cot x+C`

C

`C-cotx+cot^(-1)x`

D

`-tan^(-1)x-("cosec x")/(sec x)+C`

Text Solution

Verified by Experts

The correct Answer is:
C
`l=int(x^(2)+cos^(2)x)/(x^(2)+1)."cosec"^(2)xdx`
`=int(x^(2)+1+cos^(2)x-1)/(x^(2)+1)."cosec"^(2)xdx`
`=int(1-(sin^(2)x)/(x^(2)+1))."cosec"^(2)xdx`
`=int("cosec"^(2)x-(1)/(x^(2)+1))dx`
`=-cotx - tan^(-1)x+C_(1)=-cot x+cot^(-1)x-(pi)/(2)+C_(1)`
`=-cotx+cot^(-1)x+C" where, "C=C_(1)-(pi)/(2)`
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