Skip to main content
Mathematics LibreTexts

Table of Integrals

  1. Last updated
  2. Save as PDF
  • Page ID
    10816

    • This page is a draft and is under active development. 

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vectorC}[1]{\textbf{#1}} \)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)

    Integration Rules

    \( \int (A\textcolor{blue}{f(x)} + B\textcolor{brown}{g(x)} dx = A \int \textcolor{blue}{f(x)}dx +B \int \textcolor{brown}{g(x)} dx \)
    \( \int \textcolor{ blue}{f'(} \textcolor{ brown}{g(x)} \textcolor{ blue}{)} \textcolor{ brown}{g'(x)}dx = \textcolor{ blue}{f(} \textcolor{ brown}{g(x)} \textcolor{ blue}{)} + C \)
    \( \int \textcolor{blue}{U(x)} \textcolor{brown}{dV(x)} = \textcolor{ blue}{U(x)} \textcolor{ brown}{V(x)} - \int \textcolor{ brown}{V(x)} \textcolor{ blue}{dU(x)} \)
    \( \int_ \textcolor{ orange}{a}^ \textcolor{ magenta}{b} \textcolor{ blue}{f'(x)}dx = \textcolor{ blue}{f(} \textcolor{ magenta}{b} \textcolor{ blue}{)}- \textcolor{ blue}{f(} \textcolor{ orange}{a} \textcolor{ blue}{)} \)
    \( \displaystyle \frac {d} {dx} \int_ \textcolor{ orange}{a}^ \textcolor{ magenta}{x} \textcolor{ blue}{f(t)}dt = \textcolor{ blue}{f(} \textcolor{ magenta}{x} \textcolor{ blue}{)} \)

    Integrals for Elementary Trancendental Functions

    \( \int \textcolor{orange}{x}^\textcolor{magenta}{n}dx = \displaystyle \frac {1} {\textcolor{magenta}{n}+1} \textcolor{orange}{x}^{\textcolor{magenta}{n}+1} + C \, \), if \( \textcolor{magenta}{n} \ne -1 \)
    \( \int \displaystyle \frac {dx}{\textcolor{orange}{x}} = \ln|\textcolor{orange}{x}| + C \)
    \( \int e^\textcolor{orange}{x}dx = e^\textcolor{orange}{x} + C \)
    \( \int \textcolor{magenta}{a}^\textcolor{orange}{x} = \displaystyle \frac {\textcolor{magenta}{a}^\textcolor{orange}{x}} {\ln(\textcolor{magenta}{a})} + C \)
    \( \int \sin(\textcolor{orange}{x})dx = -\cos(\textcolor{orange}{x}) + C \)
    \( \int \cos(\textcolor{orange}{x})dx = \sin(\textcolor{orange}{x}) + C \)
    \( \int \sec^2(\textcolor{orange}{x})dx = \tan(\textcolor{orange}{x}) + C \)
    \( \int \csc^2(\textcolor{orange}{x})dx = -\cot(\textcolor{orange}{x}) + C \)
    \( \int \sec(\textcolor{orange}{x})\tan(\textcolor{orange}{x})dx = \sec(\textcolor{orange}{x}) + C \)
    \( \int \csc( \textcolor{ orange}{x})\cot( \textcolor{ orange}{x})dx = -\csc( \textcolor{ orange}{x}) + C \)
    \( \int \tan( \textcolor{orange}{x})dx = \ln|\sec( \textcolor{ orange}{x})| + C \)
    \( \int \cot( \textcolor{ orange}{x})dx = \ln|\sin( \textcolor{ orange}{x})| + C \)
    \( \int \sec( \textcolor{ orange}{x})dx = \ln|\sec( \textcolor{ orange}{x}) + \tan( \textcolor{ orange}{x})| + C \)
    \( \int \csc( \textcolor{ orange}{x})dx = \ln|\csc( \textcolor{ orange}{x}) - \cot( \textcolor{ orange}{x})| + C \)
    \( \int \displaystyle \frac {dx} {\sqrt{ \textcolor{ magenta}{a}^2- \textcolor{ orange}{x}^2}} = \sin^{-1} \left( \displaystyle \frac { \textcolor{ orange}{x}} { \textcolor{ magenta}{a}} \right)+ C \, ( \textcolor{ magenta}{a} > 0,\, | \textcolor{ orange}{x}| < \textcolor{ magenta}{a}) \)
    \( \int \displaystyle \frac {dx} { \textcolor{ magenta}{a}^2+ \textcolor{ orange}{x}^2} = \displaystyle \frac {1} { \textcolor{ magenta}{a}} \tan^{-1} \left( \displaystyle \frac { \textcolor{ orange}{x}} { \textcolor{ magenta}{a}} \right) + C \, ( \textcolor{ magenta}{a} > 0) \)
    \( \int \displaystyle \frac {dx} { \textcolor{ magenta}{a}^2- \textcolor{ orange}{x}^2} = \displaystyle \frac {1} { 2 \textcolor{ magenta}{a}} \ln \left| \displaystyle \frac { \textcolor{ orange}{x}+ \textcolor{ magenta}{a}} { \textcolor{ orange}{x}- \textcolor{ magenta}{a}} \right| + C \, ( \textcolor{ magenta}{a}>0) \)
    \( \int \displaystyle \frac {dx} { \textcolor{ orange}{x} \sqrt{ \textcolor{ orange}{x}^2- \textcolor{ magenta}{a}^2}} = \displaystyle \frac {1} { \textcolor{ magenta}{a}} \sec^{-1} \left| \displaystyle \frac { \textcolor{ orange}{x}} { \textcolor{ magenta}{a}} \right| + C \, ( \textcolor{ magenta}{a}>0,\, | \textcolor{ orange}{x}|> \textcolor{ magenta}{a}) \)

    Table of Integrals is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

    1. Back to top
    • Was this article helpful?

    Recommended articles

    1. Article type
      Section or Page
      Print CSS
      Dense
      License
      CC BY-NC-SA
      Show Page TOC
      yes
      Stage
      Draft
    2. Tags
      1. calcplot:yes
      2. jupyter:python