The value of `sqrt(2)int(sinx)/(sin(x-(pi)/(4)))dx` , is

Options

A. `x-log|cos(x-(pi)/4)|+C`
B. `x+log|x-(pi)/4|+C`
C. `x-log|sin(x-(pi)/4)|+C`
D. `x+log|sin(x-(pi)/4)|+C`

Answer

D

Explanation

Let `l=sqrt(2)int(sinx)/(sin(x+(pi)/4))dx`
Put `x-(pi)/4=timpliesdx=dt`
`:.l=sqrt(2)int(sin((pi)/4+t)dt)/(sint)`
`=sqrt(2)int("sin"(pi)/4cost+"cos"(pi)/4sint)/(sint)dt`
`=sqrt(2)int(1/(sqrt(2))(cos)/(sint)+1/(sqrt(2)))dt`
`=int(cot t+1)dt`
`=log|sint|+t+C_(1)`
`=x+log|sin(x-(pi)/4)|+C`
`[ :' c_(1)-(pi)/4=c]`