\int{basic syntax}
Basic integral syntax
The \int command is available in LaTeX's core math mode, but loading amsmath is strongly recommended — it corrects spacing, adds multi-integral commands, and provides the \, thin space typically written before dx.
\documentclass{article}
\usepackage{amsmath}
\begin{document}
% Indefinite integral
\[ \int f(x)\,dx \]
% Definite integral
\[ \int_{a}^{b} f(x)\,dx \]
% Definite integral evaluated
\[ \int_{0}^{1} x^{2}\,dx = \frac{x^{3}}{3}\bigg|_{0}^{1} = \frac{1}{3} \]
% Common antiderivatives
\[ \int \frac{1}{1+x^{2}}\,dx = \arctan(x) + C \]
\[ \int e^{x}\,dx = e^{x} + C \]
\[ \int \frac{1}{x}\,dx = \ln|x| + C \]
\end{document}No limits specified. The result is a family of antiderivatives differing by a constant C.
Lower limit a and upper limit b are placed below and above the integral sign respectively.
Use \infty from amsmath for infinite bounds. LaTeX renders the infinity symbol automatically.
Use \, before dx for a thin space (standard typographic convention in display math).
\section{Definite vs Indefinite}
Definite vs indefinite integrals
The only syntactic difference is whether you supply the lower _{a} and upper ^{b} limits. LaTeX places them below and above the integral symbol in display mode.
Indefinite
\int f(x)\,dx
No bounds. Returns a function plus an arbitrary constant C. Used when finding antiderivatives.
Definite
\int_{a}^{b} f(x)\,dx
Evaluated between a and b. Returns a numerical value equal to the signed area under the curve.
Improper (infinite bounds)
\int_{0}^{\infty} f(x)\,dx
One or both bounds are ±∞. Use \infty from amsmath. Convergence must be checked separately.
Evaluation bar
\bigg|_{0}^{1}
Typeset the evaluation bar after an antiderivative with \bigg| and the limits as a subscript/superscript pair.
\iint{multiple integration}
Multiple integration
The amsmath package provides \iint (double) and \iiint (triple) with correctly spaced integral signs. For closed surface integrals, add the esint package which supplies \oiint and \oiiint.
\documentclass{article}
\usepackage{amsmath}
\usepackage{esint} % oiint, oiiint
\begin{document}
% Double integral over domain D
\[ \iint_{D} f(x,y)\,dx\,dy \]
% Triple integral over volume V
\[ \iiint_{V} \rho(x,y,z)\,dx\,dy\,dz \]
% Line integral (closed path)
\[ \oint_{C} \mathbf{F} \cdot d\mathbf{r} \]
% Closed surface integral (requires esint)
\[ \oiint_{S} \mathbf{F} \cdot d\mathbf{S} \]
% Gaussian integral
\[ \int_{-\infty}^{+\infty} e^{-x^{2}}\,dx = \sqrt{\pi} \]
% Physics: Boltzmann partition function
\[ Z = \int_{-\infty}^{\infty} e^{-\beta H(q,p)}\,dq\,dp \]
\end{document}\oint{contour integrals}
Contour integrals
Closed-path integrals use a circle overlaid on the integral sign to indicate integration around a closed curve or surface. All contour integral commands accept the same subscript and superscript syntax as \int.
Line integral along a closed curve. Core amsmath command — no extra package needed.
Closed surface integral (double). Requires \usepackage{esint}. Used in Gauss's law.
Closed volume integral (triple). Requires esint. Used for 3D divergence theorem applications.
Explicitly clockwise contour integral. Useful in complex analysis to specify orientation.
Counter-clockwise contour integral. The positive (standard) orientation in complex analysis.
Variant clockwise contour. Alternate glyph provided by esint for style preference.
\limits{placement reference}
Limits and bounds placement
LaTeX automatically chooses limit placement based on context: above/below in display mode, to the side in inline mode. You can override this with \limits and \nolimits.
| Syntax | Placement |
|---|---|
| \int_{a}^{b} | Subscript a below, superscript b above |
| \int\limits_{a}^{b} | Forces limits above/below even in inline mode |
| \int\nolimits_{a}^{b} | Forces limits to the side (inline style) even in display mode |
| \displaystyle\int_{a}^{b} | Display-size integral with limits in paragraph text |
\section{Physics integrals}
Physics integrals
Physics equations frequently involve complex integral expressions. Here are three canonical examples with their full LaTeX syntax.
Maxwell's Equations (Integral form)
\oint_{S} \mathbf{E} \cdot d\mathbf{S} = \frac{Q_{\text{enc}}}{\varepsilon_0}
Gauss's law for electric flux through a closed surface. Requires \oiint from esint for a double closed surface integral.
Boltzmann Partition Function
Z = \int_{-\infty}^{\infty} e^{-\beta H(q,p)}\,dq\,dp
Statistical mechanics partition function integrated over all phase-space coordinates q and p.
Fourier Transform
\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\,e^{-2\pi i x \xi}\,dx
Transforms a time-domain signal into the frequency domain. The exponential kernel uses inline math mode for the exponent.
Classic Integrals — live compilation
The playground is pre-loaded with Gaussian, Dirichlet, Fresnel, and Gauss' theorem examples. Compile instantly with pdfLaTeX.
\section{FAQ}