Evaluate the following indefinite integrals :
`(i) int8cosxcos^(3)3xdx`
`(ii) int(x+2005)/((x+2007)^(2))dx`
`(iii) int(x+5)/(sqrt(x-3))dx`

Explanation

`(i)` We have ,
`8cosxcos^(3)3x=(2cosx)(4cos^(3)3x)`
`=(2cosx)(cos9x+3cos3x)`
`=cos10x+cos8x+3cos2x+3cos4x`
`:.int8cosxcos^(3)3xdx=int(cos10x+cos8x+3cos2x+3cos4x)dx`
`=(sin10x)/(10)+(sin8x)/(8)+(3sin2x)/(2)+(3sin4x)/(4)+C`, where `C` is a constant of integration
`(ii)` We have ,
`int(x+2005)/((x+2007)^(2))dx=int(x+2007-2)/((x+2007)^(2))dx`
`=int(1)/((x+2007))dx-2int(1)/((x+2007)^(2))dx`
`=ln|x+2007|+(2)/(x+2007)+C`, where is `C` is constant of integration.