List of basic formulas for integrating fractions:

  1. Integral of a fraction of the form ( \frac{1}{x} ):
    [
    \int \frac{1}{x} \, dx = \ln |x| + C
    ]
  2. Integral from a fraction of type ( \frac{1}{ax + b} ):
    [
    \int \frac{1}{ax + b} \, dx = \frac{1}{a} \ln |ax + b| + C
    ]
  3. 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
    ]
  4. 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
    ]
  5. 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
    ]
  6. 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.

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