I have an improper fraction that looks like this: $\int \frac{x^2+1}{x^2-5x+6}dx$

I wanted to know what's the best way to simplify this?

For example, an improper fraction is given when $n \ge m$. In the book I'm currently reading, it mentions to work out the integral by always dividing the numerator by the denominator, and obtaining a polynomial plus a proper fraction.

The result should be this: $$\int 1 + \frac{5x-5}{x^2-5x+6}dx$$

However, given the information above I cannot seem to find this answer. Expanding this out it gives the same integral, though what are the steps to achieve an integral in this form?

## Best Answer

One way to approach this particular problem is as follows:

$$\frac{x^2+1}{x^2-5x+6} = \frac{x^2-5x+5x-5+5+1}{x^2-5x+6}=\frac{(x^2-5x+6)+(5x-5)}{x^2-5x+6} = 1 + \frac{5x-5}{x^2-5x+6}$$