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. |
