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int(dx)/(cos x-sinx) is equal to...

`int(dx)/(cos x-sinx)` is equal to

A

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

B

`(1)/(sqrt2)log|cot((x)/(2))|+C`

C

`(1)/(sqrt2)log|tan((x)/(2)-(3pi)/(8))|+C`

D

`(1)/(sqrt2)log|tan((x)/(2)+(3pi)/(8))|+C`

Text Solution

Verified by Experts

The correct Answer is:
D
`l=int(dx)/(cosx-sinx)`
`=(1)/(sqrt2)int(dx)/(((1)/(sqrt2)cosx-(1)/(sqrt2)sinx))`
`=(1)/(sqrt2)int(dx)/(cos(x+(pi)/(4)))`
`=(1)/(sqrt2)intsec(x+(pi)/(4))dx`
`=(1)/(sqrt2)log|tan((pi)/(4)+(x)/(2)+(pi)/(8))|+C`
`=(1)/(sqrt2)log|tan((x)/(2)+(3pi)/(8))|+C`
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