### Integral Calculus

**Integration Formulas:**

Formula | Description |
---|---|

Power rule | $$\int x^n dx = \frac{x^{n+1}}{n+1} + C,$$ |

Logarithmic rule | $$\int \frac{1}{x} dx = \ln |

Exponential rule | $$\int e^x dx = e^x + C.$$ |

Trigonometric rule | |

- $$\int \sin x dx = -\cos x + C.$$ | |

- $$\int \cos x dx = \sin x + C.$$ | |

- $$\int \tan x dx = \ln | \sec x |

- $$\int \csc x dx = -\ln | \csc x + \cot x |

- $$\int \sec x dx = \ln | \sec x + \tan x |

Integration by partials | $$\int udv = uv - \int vdu,$$ where $u$ and $v$ are functions of $x$ and $du$ and $dv$ are their respective differentials. |

Partial fractions | Used for integrating rational functions. |

Improper integrals | Integrals that do not converge absolutely. |

Beta function | $$\int_0^1 x^{p-1}(1-x)^{q-1} dx,$$ where $p$ and $q$ are positive real numbers. |

Gamma function | $$\Gamma(z) = \int_0^\infty e^{-t}t^{z-1} dt,$$ where $z$ is a complex number. |

Riemann integral | Provides a rigorous definition of the integral. |

Definite integrals | Integrals with both upper and lower limits of integration. |

Integration techniques | |

- U-substitution | Substituting a new variable $u = g(x)$ to simplify the integral. |

- Integration by trigonometric substitution | Using trigonometric identities to simplify the integral. |

- Integration by rationalization | Rewriting the integrand so that the denominator can be factored into a product of linear factors. |

- Integration by tabular methods | Using a table of integrals to find the value of the integral. |