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If int(sinx)/(sin(x-(pi)/(4)))dx=Af(x)+(...

If `int(sinx)/(sin(x-(pi)/(4)))dx=Af(x)+(1)/(sqrt2)log[|sinx-cosx|]+c,` then

A

`A=(1)/(sqrt2),f(x)=sinx`

B

`A=sqrt2,f(x)=cosx`

C

`A=sqrt2,f(x)=x`

D

`A=(1)/(sqrt2),f(x)=x`

Text Solution

Verified by Experts

The correct Answer is:
D
`int(sinx)/(sin(x-(pi)/(4)))dx`
`=int(sin(u+(pi)/(4)))/(sinu)du" "("put "u=x-(pi)/(4))`
`=int(sin ucos.(pi)/(4)+cosu sin.(pi)/(4))/(sinu)du`
`=(1)/(sqrt2)u+(1)/(sqrt2)log|sinu|+c`
`=(1)/(sqrt2)x+(1)/(sqrt2)log(|sinx-cosx|)+c`
`therefore" "A=(1)/(sqrt2),f(x)=x`
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