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int(dx)/(sinx-cosx+sqrt2) is equal to...

`int(dx)/(sinx-cosx+sqrt2)` is equal to

A

`-(1)/(sqrt2)tan((x)/(2)+(pi)/(8))+C`

B

`(1)/(sqrt2)tan((x)/(2)+(pi)/(8))+C`

C

`(1)/(sqrt2)cot((x)/(2)+(pi)/(8))+C`

D

`-(1)/(sqrt2)cot((x)/(2)+(pi)/(8))+C`

Text Solution

Verified by Experts

The correct Answer is:
D
`int(dx)/(sinx-cosx+sqrt2)`
`=int(dx)/(sqrt2((1)/(sqrt2)sinx-(1)/(sqrt2)cosx)+sqrt2)`
`=(1)/(sqrt2)int(dx)/(1-cos (x+(pi)/(4)))=(1)/(sqrt2)int(dx)/(2sin^(2)((x)/(2)+(pi)/(8)))`
`=(1)/(2sqrt2)int(dx)/(sin^(2)((x)/(2)+(pi)/(8)))=(1)/(2sqrt2)int"cosec"^(2)((x)/(2)+(pi)/(8))dx`
`=(1)/(2sqrt2)[(-cot ((x)/(2)+(pi)/(8)))/((1)/(2))]+C=-(1)/(sqrt2)cot((x)/(2)+(pi)/(8))+C`
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