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