How Do You Integrate \(\int \frac{x-3}{x^2+2x-5}dx\) with Complex Roots?

[Tasy]

$\int \frac{x-3}{x^2+2x-5}dx$

How integrate this task?

$x^2+2x-5=0$

$D=24$, so I can't get real root.

 

[Curious3141]

$\int \frac{x-3}{x^2+2x-5}dx$

How integrate this task?

$x^2+2x-5=0$

$D=24$, so I can't get real root.

OK, that line with the discriminant made no sense. The discriminant is greater than zero, so clearly the quadratic does have real, distinct roots.

Hint: start off with completing the square on the denominator. In other words, express the denominator in the form $(x+a)^2 + c$ and go from there.