Question

How do you integrate `int (5x^2+3x-2)/(x^3+2x^2)` using partial fractions?

Answer 1

`1/x+ln|x^2(x+2)^3|+C_1`

Explanation:

The `Dr.=x^3+2x^2=x^2(x+2)`

So, we are having linear repeated factors. In such case, we take,

`(5x^2+3x-2)/(x^3+2x^2)=A/x+B/x^2+C/(x+2)....(1), where, A,B,C in RR`

`=(Ax(x+2)+B(x+2)+Cx^2)/(x^2(x+2))`

`rArr 5x^2+3x-2=Ax(x+2)+B(x+2)+Cx^2..............(2)`

Eventhough `(2)` is true `AAx in RR`, we select,

`x=0 rArr -2=2B rArr B=-1`

`x=-2 rArr20-6-2=12=4C rArr C=3`

`x=1 rArr 5+3-2=6=3A+3B+C=3A-3+3=3A`

`rArrA=2`

`:. I=int(5x^2+3x-2)/(x^3+2x^2)dx`

`=int2/xdx-int1/x^2dx+int3/(x+2)dx`

`=2lnx+1/x+3ln|x+2|`

`=1/x+ln|x^2(x+2)^3|+C_1`

Hope, this will be of a little Help! Enjoy Maths. & spread the Joy!