Question

How do I find `int_0^3dx/(x^2-6x+5)`?

Answer 1

This definite integral does not converge (it has an infinite answer).

Normally for integrals like this, a Partial Fractions approach would be useful. To begin, remember that rational equations such as:

`a/x + b/(x+1)`

can be combined by cross multiplying:

`(a(x+1) + b(x))/(x(x+1))` = `(ax + a + bx) / (x^2 + x)`

The Partial Fractions approach aims to reverse this process in order to turn certain rational integrals into a sum of simpler equations. Often this can result in a sum of simple linear equations which can easily be solved with Substitution.

The first step for this type of problem would be to try and factor the denominator as much as possible:

`1/(x^2 - 6x + 5) => 1/((x-1)(x-5))`

Ordinarily you would continue with the Partial Fractions approach, but notice the denominator. You have two factors which could potentially be equal to zero, which would result in an infinite answer (division by zero). By setting the 2 different equations equal to 0, you can see that this occurs at `x=1` and `x=5`. Since our limits of integration range from `0` to `3`, the integral would contain a section that stretches to infinity, causing our integral to also become infinite:

graph{1 / (x^2 - 6x + 5) [-10, 10, -5, 5]}
(You can see that from 0 to 3, the line hits a critical point which causes an infinite integral.)

If this were not the case, the Partial Fractions method would have turned this integral into a simple one solvable by substitution.