# Integration by partial fractions- a quadratic

• TG3
In summary, the conversation discusses how to integrate the expression (x-1) / (x^2 - 4x + 5) and the approach to take if the quadratic equation does not have real roots. The technique of completing the square in the denominator is suggested.

#### TG3

Homework Statement

Integrate: (x-1) / (X^2 - 4x +5)

The attempt at a solution

Normally I would try to factor this into something like (x-1) (x+3) (That's an example completely unrelated to this problem.)
However, as no easy factors quickly occurred to me I did a run through of the quadratic equation, and got 4 +or- (16- 4x1x5)^.5 / 2 = 4 + (-4)^.5 / 2

At my basic level of Calculus, imaginary numbers are taboo, so I want to avoid the square root of a negative number.

Am I missing an obvious factoring, am I performing the quadratic wrong (that would be embarrassing) or should I try another technique? And if another technique... what is it?

well if x^2 - 4x +5=0 has no real roots, then you need to express the partial fraction in the form (ax+b)/(Ax^2+Bx+C)

The technique you are missing is to complete the square in the denominator, x^2-4x+5=(x-2)^2+1. Substitute u=x-2. Can you see how to go from there?

rock.freak667 said:
well if x^2 - 4x +5=0 has no real roots, then you need to express the partial fraction in the form (ax+b)/(Ax^2+Bx+C)

Quoi? That's the same form as the original integrand!?? It's no partial fraction at all.