Factor to solve partial fractions decompositions

Question

$\int$$ (x^2-x-21)dx\over(2x^3-x^2+8x-4))$

I know how to factor the denominator but I don't know how to factor the numerator. Could I get a step by step breakdown of how to solve??

Answer 1

HINT:

Why do you want to factorize the numerator?

As $$2x^3-x^2+8x-4=x^2(2x-1)+4(2x-1)=(2x-1)(x^2+4)$$

Using Partial Fraction Decomposition formula, $$\frac{x^2-x-21}{2x^3-x^2+8x-4}=\frac A{2x-1}+\frac{Bx+C}{x^2+4}$$

$$\implies x^2-x-21=A(x^2+4)+(Bx+C)(2x-1)$$

Now either compare the coefficients of the different powers of $x$

or set $2x-1=0,x^2+4=0\implies x=\pm2i$

to find $A,B,C$