Evaluate: `int(dx)/(x(x^5+3))`

Explanation

`Let I=int(dx)/(x(x^(5)+3))=int(x^(4))/(x^(5)(x^(5)+3))dx.`
Putting `x^(5)=t and 5x^(4)dx=dt`in (i) ,we get
`I=(1)/(5)int(dt)/(t(t+3)).`
`Let (1)/(t(t+3))=(A)/(t)+(B)/(t+3).`
Then ,`1-= A(t+3)+Bt.`
Putting `t=0` on both sides of (iii) , we get `A=(1)/(3).`
Putting `t=-3` on both sides of (iii) , we get `B=(-1)/(3).`
`therefore (1)/(t(t+3))=(1)/(3t)-(1)/(3(t+3))`
`implies I=(1)/(5)int{(1)/(3t)-(1)/(3(t+3))}dt`
`=(1)/(15)int(1)/(t)dt-(1)/(15)int(1)/((t+3))+C`
`=(1)/(15)log|x^(5)|-(1)/(15)log|x^(5)+3|+C.`