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|cos(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=sqrt2 int(sinx)/(sin(x-(pi)/(4)))dx`
Put `x-(pi)/(4)=t rArr dx=dt`
`therefore" "l=sqrt2 int(sin((pi)/(4)+t)dt)/(sint)`
`=sqrt2 int(sin .(pi)/(4)cost+cos.(pi)/(4)sint)/(sint)dt`
`=sqrt2 int((1)/(sqrt2)(cost)/(sint)+(1)/(sqrt2))dt`
`=int(cot +1)dt=log|sint|+t+C_(1)`
`=x +log|sin(x-(pi)/(4))|+C" "[becauseC_(1)-(pi)/(4)=C]`