`(cosx-sinx)/(sqrt(8-sin2x))`

Explanation

`"Let I"int (cos x-sinx)/(sqrt((8-sin2x)))dx`
`"Let I"=int (cos x-sin x)/(sqrt(8-sin2x))dx`
Put `sinx+cos x=1`
`therefore (cos x-sinx)dx=dt`
Also, `(sin x+cos x)^(2)=t^(2)`
`therefore sin^(2)x+cos^(2)x+2sinxcosx=t^(2)`
`therefore 1+sin2x=t^(2)" "therefore 2x=t^(2)-1`
`therefore I=int (1)/(sqrt(8-(t^(2)-1)))dt =int (1)/(sqrt((8-t^(2)+1)))dt`
`=int (1)/(sqrt(3^(2)-t^(2)))dt`
`=sin^(-1)((t)/(3))+c=sin^(-1)((sin x+cos x)/(3))+c`