Evaluating an indefinite integral by u-substitution

Let $u = x^2 - 3x + 7$. Then $du = (2x-3)\,dx$, which is exactly the numerator times $dx$. So the integral becomes $\int \frac{du}{u^2} = \int u^{-2}\, du$.

Integrating gives $\int u^{-2}\, du = -u^{-1} + C = -\frac{1}{u} + C$. Substituting back $u = x^2-3x+7$ gives the final answer $-\dfrac{1}{x^2-3x+7} + C$.