Question

How do you integrate `int (x^3+x^2+x+2)/(x^4+x^2)` using partial fractions?

How do you integrate `int (x^3+x^2+x+2)/(x^4+x^2)dx` using partial fractions?

Answer 1

`ln|x|-2/x-arc tanx+C`

Explanation:

Let `I=int(x^3+x^2+x+2)/(x^4+x^2)dx=int(x^3+x^2+x+2)/(x^2(x^2+1))dx`

On the first hand inspection, we find that we have no go but to use

the Method of Partial Fraction to decompose the Integrand, but,

`"The Nr."`

`=(x^3+x^2+x+2)=ul(x^3+x)+ul(x^2+1)+1`

`=x(x^2+1)+1(x^2+1)+1=(x^2+1)(x+1)+1`.

Hence, `(x^3+x^2+x+2)/(x^2(x^2+1))={(x^2+1)(x+1)+1}/(x^2(x^2+1))`

`={(x^2+1)(x+1)}/ (x^2(x^2+1))+1/(x^2(x^2+1))`

`=((x^2+1)(x+1))/(x^2(x^2+1))+1/(x^2(x^2+1))`

`=(x+1)/x^2+{(x^2+1)-x^2)/(x^2(x^2+1)`

`=x/x^2+1/x^2+(x^2+1)/(x^2(x^2+1))-x^2/(x^2(x^2+1)`

`=1/x+1/x^2+1/x^2-1/(x^2+1)`.

Therefore,

`I=int[1/x+2/x^2-1/(x^2+1)]dx`

`=ln|x|-2/x-arc tanx+C`

Enjoy maths.!