Evaluate `int (x^(2))/((1+x^(3))(2+x^(3)))dx.`

Explanation

Putting `x^(3)=t and x^(2)dx=(1)/(3)dt,`we get
`I=int(x^(2))/((1+x^(3))(2+x^(3)))dx=(1)/(3).int(dt)/((1+t)(2+t)).`
`Let (1)/((1+t)(2+t))=(A)/((1+t))+(B)/((2+t)).`then,
`1-=A(2+t)+B(1+t).`
Putting `t=-1` in (i) , we get `A=1.`
Putting `t=-2`in (i) , we get `B=-1.`
`therefore (1)/((1+t)(2+t))=(1)/((1+t))-(1)/((2+t))`
`implies I=int (dt)/((1+t)(2+t))=int(dt)/((1+t))-int(dt)/((2+t))`
`=log |1+t|-log|2+t|+C`
`=log |(1+t)/(2+t)|+C`
`=log |(1+x^(3))/(2+x^(3))|+C`