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

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

Discussion

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