Notes on integration using fractions

Integrating fractions often involves recognizing specific forms. For $\frac{1}{x}$, the integral is $\ln|x| + C$. For other powers like $\frac{1}{x^n}$ (where $n \neq 1$), rewrite it as $x^{-n}$ and apply the power rule:

$$\int \frac{1}{x^n} \,dx = \int x^{-n} \,dx = \frac{x^{-n+1}}{-n+1} + C$$

When the numerator is the derivative of the denominator, u-substitution is effective. For more complex rational functions, partial fraction decomposition simplifies the integrand into terms that are easier to integrate, often resulting in logarithmic expressions.