Find following integrals
`(i) int(x^(5)-2)/(x^(4))dx`
`(ii) int{x^((3)/(2))+4e^(x)+(1)/(x)}dx`
`(iii) int cosecx{cosecx-cotx}dx`
`(iv) int(1+sinx)/(cos^(2)x)dx`

Explanation

`(i) int(x^(5)-2)/(x^(4))dx=intxdx-int(2)/(x^(4))dx` [Property 4]
`=(x^(1+1))/(1+1)+C_(1)-2{(x^(-4+1))/(-4+1)+C_(2)}`, `C_(1)`, `C_(2)` are constants of inetgration
`=(x^(2))/(2)-2.(x^(-3))/(-3)+(-2C_(2))+C_(1)`
`=(x^(2))/(2)+(2)/(3x^(3))+C`, where `C=C_(1)-2C_(2)` is another constant of integration.
`(ii) int{x^((3)/(2))+4e^(x)+(1)/(x)}dx=intx^((3)/(2))dx+4inte^(x)dx+int(1)/(x)dx` [Property 4]
`={(x^((3)/(2))+1)/((3)/(2)+1)+C_(1)}+4{e^(x)+C_(2)}+log|x|+C_(3)`
`=(2)/(5)x^((5)/(2))+4e^(x)+log|x|+C`, {where `C=C_(1)+4C_(2)+C_(3)`}
i.e., constant of integration
Note : From now onwards, we shall write only one constant of integration in final answer.
`(iii) intcosecx(cosecx-cotx)dx=intcosec^(2)xdx=intcosecxcotxdx`
`=-cotx+cosecx+C`
`(iv) (1+sinx)/(cos^(2)x)dx=int(1)/(cos^(2))dx+int(sinx)/(cos^(2)x)dx`
`=intsec^(2)xdx+intsecxtanxdx`
`=tanx+secx+C`