Here is a list of some formulas and methods that can be useful when integrating irrational functions:

  1. Integral of the root:
    [
    \int \sqrt{x} \, dx = \frac{2}{3} x^{3/2} + C
    ]
  2. Integral of a root with a linear factor:
    [
    \int \sqrt{ax + b} \, dx = \frac{2}{3a} (ax + b)^{3/2} + C
    ]
  3. Integral of a fraction with a root:
    [
    \int \frac{1}{\sqrt{x}} \, dx = 2\sqrt{x} + C
    ]
  4. Интеграл от функции вида (\sqrt{a^2 – x^2}):
    [
    \int \sqrt{a^2 – x^2} \, dx = \frac{x}{2} \sqrt{a^2 – x^2} + \frac{a^2}{2} \arcsin\left(\frac{x}{a}\right) + C
    ]
  5. Интеграл от функции вида (\sqrt{x^2 + a^2}):
    [
    \int \sqrt{x^2 + a^2} \, dx = \frac{x}{2} \sqrt{x^2 + a^2} + \frac{a^2}{2} \ln\left| x + \sqrt{x^2 + a^2} \right| + C
    ]
  6. Интеграл от функции вида (\sqrt{x^2 – a^2}):
    [
    \int \sqrt{x^2 – a^2} \, dx = \frac{x}{2} \sqrt{x^2 – a^2} – \frac{a^2}{2} \ln\left| x + \sqrt{x^2 – a^2} \right| + C
    ]
  7. Substitution method: If the function has a complex form, you can use the substitution (u = g(x)), where (g(x)) is a function that simplifies the integral.
  8. Integration by parts: If a function can be represented as a product of two functions, the integration by parts formula can be used:
    [
    \int u \, dv = uv – \int v \, du
    ]

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