Question

How do you integrate `(4x^2-x+3)/((x+5)(x-1)(x-2))` using partial fractions?

Answer 1

`-ln|x-1|+17/7ln|x-2|+18/7ln|x+5|+K`, or,

`ln|((x+5)^(18/7)(x-2)^(17/7))/(x-1)|+K`.

Explanation:

Let `I=int(4x^2-x+3)/((x+5)(x-1)(x-2))dx`.

We will use the Method of Partial Fraction to split the Integrand

`(4x^2-x+3)/((x+5)(x-1)(x-2))=A/(x-1)+B/(x-2)+C/(x+5);`

where, `A,B,C in RR`.

To determine, `A,B,C`, let us use Heavyside's Cover-up Method :-

`A=[(4x^2-x+3)/((x+5)(x-2))]_(x=1)=6/-6=-1`;

`B=[(4x^2-x+3)/((x+5)(x-1))]_(x=2)=17/7`;

`C=[(4x^2-x+3)/((x-1)(x-2))]_(x=-5)=108/((-6)(-7))=18/7`.

Therefore, `I=int[-1/(x-1)+(17/7)/(x-2)+(18/7)/(x+5)]dx`

`=-ln|x-1|+17/7ln|x-2|+18/7ln|x+5|+K`, or,

`=ln|((x+5)^(18/7)(x-2)^(17/7))/(x-1)|+K`.

Enjoy Maths.!