List of basic formulas for integrating fractions:
- Integral of a fraction of the form ( \frac{1}{x} ):
[
\int \frac{1}{x} \, dx = \ln |x| + C
] - Integral from a fraction of type ( \frac{1}{ax + b} ):
[
\int \frac{1}{ax + b} \, dx = \frac{1}{a} \ln |ax + b| + C
] - Integral of a fraction of the form ( \frac{1}{x^2 + a^2} ):
[
\int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a} \tan^{-1} \left( \frac{x}{a} \right) + C
] - Интеграл от дроби вида ( \frac{1}{\sqrt{a^2 – x^2}} ):
[
\int \frac{1}{\sqrt{a^2 – x^2}} \, dx = \sin^{-1} \left( \frac{x}{a} \right) + C
] - Интеграл от дроби вида ( \frac{1}{x^2 – a^2} ):
[
\int \frac{1}{x^2 – a^2} \, dx = \frac{1}{2a} \ln \left| \frac{x – a}{x + a} \right| + C
] - Integral of a fraction of the form ( \frac{P(x)}{Q(x)} ) (where ( P(x) ) and ( Q(x) ) are polynomials):
- If the degree ( P(x) < ) the degree ( Q(x) ), one can use the polynomial division method and then integrate the remainder.
- If the power of ( P(x) \geq ) is the power of ( Q(x) ), the division is performed first and then the integration.