Evaluate: (i) `int(1+cosx)/((x+sinx)^3)\ dx` (ii) `int(cosx-sinx)/(1+sin2x)\ dx`

Explanation

`(i) int(1+cosx)/((x+sinx)^3)\ dx`
taking `x+sinx=t`
`=>1+cosx dx =dt`
so,` int (dt)/t^3`
`=-1/(2t^2)+c`
`=-1/(2(x+sinx)^2)+c`
(ii) `int(cosx-sinx)/(1+sin2x)\ dx`
`int (cosx-sinx)/(cos^2x+sin^2x+2sinx*cosx )dx`
`=int (cosx-sinx)/(cosx+sinx)^2 dx`
taking `(cosx+sinx)=t`
`=>-sinx+cosx dx =dt`
so, `int 1/t^2 dt`
`=-1/t +c`
`=-1/(cosx+sinx)+c`