Appendix I: Integral Table
- Page ID
- 215888
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)To save space, only one member of each antiderivative family appears for most integrals below; for example, you should interpret \(\displaystyle \int \cos(x)\, dx = \sin(x)\) as \(\displaystyle \int \cos(x)\, dx = \sin(x) +C\), where \(C\) is an arbitrary constant.
Basic Patterns
\(\displaystyle \int k\cdot f(x) \, dx = k\cdot\int f(x)\, dx\)
\(\displaystyle \int F'(ax+b) \, dx) = \frac{1}{a}\cdot F(x)\)
\(\displaystyle \int F'(g(x))\cdot g'(x) \, dx) = F(g(x))\)
\(\displaystyle \int \left[f(x) \pm g(x)\right]\, dx = \int f(x)\, dx \pm \int g(x)\, dx\)
\(\displaystyle \int \left[f(x) \cdot g'(x)\right]\, dx = f(x)\cdot g(x) - \int f'(x)\cdot g(x) \, dx\)
Powers
\(\displaystyle \int x^p \, dx = \frac{1}{p+1} x^{p+1}\) \qquad \)(p \neq -1)\)
\(\displaystyle \int \frac{1}{x} \ dx = \ln\left|x\right|\)
Exponential Functions
\(\displaystyle \int e^x \, dx = e^x\)
\(\displaystyle \int a^x \, dx = \frac{a^x}{\ln(a)} (a >0)\)
\(\displaystyle \int x\cdot e^x \, dx = (x-1)e^x\)
\(\displaystyle \int x^2 \cdot e^x \, dx = (x^2-2x+2)e^x\)
\(\displaystyle \int x^n \cdot e^x \, dx = x^n\cdot e^x - n \int x^{n-1}e^x\, dx\)
Logarithmic Functions
\(\displaystyle \int \ln(x) \, dx = x\ln(x)-x\)
\(\displaystyle \int \ln(a^2+x^2) \, dx = x\cdot\ln(a^2+x^2)+2a\arctan\left(\frac{x}{a}\right)-2x\)
\(\displaystyle \int \frac{\ln(x)}{x} \, dx = \frac12\left[\ln(x)\right]^2\)
\(\displaystyle \int x^p \cdot \ln(x) \, dx = x^{p+1}\left[\frac{\ln(x)}{p+1}-\frac{1}{(p+1)^2}\right] \qquad (p \neq -1)\)
Trigonometric Functions
\(\displaystyle \int \sin(x) \, dx = -\cos(x)\)
\(\displaystyle \int \tan(x) \, dx = \ln\left(\left|\sec(x)\right|\right)\)
\(\displaystyle \int \sec(x) \, dx = \ln\left(\left|\sec(x)+\tan(x)\right|\right)\)
\(\displaystyle \int \cos(x) \, dx = \sin(x)\)
\(\displaystyle \int \cot(x) \, dx = \ln\left(\left|\sin(x)\right|\right)\)
\(\displaystyle \int \csc(x) \, dx = \ln\left(\left|\csc(x)-\cot(x)\right|\right)\)
\(\displaystyle \int \sin^2(x) \, dx = \frac12 x - \frac14 \sin(2x)\)
\(\displaystyle \int \sin^n(x) \, dx = -\frac{1}{n} \sin^{n-1}(x)\cos(x) + \frac{n-1}{n}\int \sin^{n-2}(x)\, dx\)
\(\displaystyle \int \cos^2(x) \, dx = \frac12 x + \frac14 \sin(2x)\)
\(\displaystyle \int \cos^n(x) \, dx = \frac{1}{n} \cos^{n-1}(x)\sin(x) + \frac{n-1}{n}\int \cos^{n-2}(x)\, dx\)
\(\displaystyle \int \sec^2(x) \, dx = \tan(x)\)
\(\displaystyle \int \sec^3(x) \, dx = \frac12\sec(x)\tan(x)+\frac12\ln\left(\left|\sec(x)+\tan(x)\right|\right)\)
\(\displaystyle \int \sec(x)\tan(x) \, dx = \sec(x)\)
\(\displaystyle \int \sec^n(x) \, dx = \frac{1}{n-1} \sec^{n-2}(x)\tan(x) + \frac{n-2}{n-1}\int \sec^{n-2}(x)\, dx\)
\(\displaystyle \int \csc^2(x) \, dx = -\cot(x)\)
\(\displaystyle \int \csc^3(x) \, dx = -\frac12 \csc(x)\cot(x) + \frac12 \ln\left(\left|\csc(x)-\cot(x)\right|\right)\)
\(\displaystyle \int \csc(x)\cot(x) \, dx = -\csc(x)\)
\(\displaystyle \int \csc^n(x) \, dx = -\frac{1}{n-1} \csc^{n-2}(x)\cot(x) + \frac{n-2}{n-1}\int \csc^{n-2}(x)\, dx\)
\(\displaystyle \int \sin(ax)\cos(bx) \, dx = -\frac{\cos\left((a-b)x\right)}{2(a-b)}-\frac{\cos\left((a+b)x\right)}{2(a+b)} \quad (a\neq \pm b)\)
\(\displaystyle \int \sin(ax)\sin(bx) \, dx = \frac{\sin\left((a-b)x\right)}{2(a-b)}-\frac{\sin\left((a+b)x\right)}{2(a+b)} \quad (a\neq \pm b)\)
\(\displaystyle \int x \sin(x) \, dx = -x\cos(x)+\sin(x)\)
\(\displaystyle \int \cos(ax)\cos(bx) \, dx = \frac{\sin\left((a-b)x\right)}{2(a-b)}+\frac{\sin\left((a+b)x\right)}{2(a+b)} \quad (a\neq \pm b)\)
\(\displaystyle \int x \cos(x) \, dx = x\sin(x)+\cos(x)\)
\(\displaystyle \int x^n \sin(x) \, dx = -x^n \cos(x) + n \int x^{n-1}\cos(x)\, dx\)
\(\displaystyle \int x^n \cos(x) \, dx = x^n \sin(x) - n \int x^{n-1}\sin(x)\, dx\)
Hyperbolic Functions
\(\displaystyle \int \sinh(x) \, dx = \cosh(x)\)
\(\displaystyle \int \tanh(x) \, dx = \ln\left(\cosh(x)\right)\)
\(\displaystyle \int \mbox{sech}(x) \, dx = \arctan(\sinh(x))\)
\(\displaystyle \int \cosh(x) \, dx = \sinh(x)\)
\(\displaystyle \int \coth(x) \, dx = \ln\left(\left|\sinh(x)\right|\right)\)
\(\displaystyle \int \mbox{csch}(x) \, dx = \ln\left(\left|\coth(x)-\mbox{csch}(x)\right|\right)\)
\(\displaystyle \int \mbox{sech}^2(x) \, dx = \tanh(x)\)
\(\displaystyle \int \mbox{sech}(x)\tanh(x) \, dx = -\mbox{sech}(x)\)
\(\displaystyle \int \mbox{csch}^2(x) \, dx = -\coth(x)\)
\(\displaystyle \int \mbox{csch}(x)\coth(x) \, dx = -\mbox{csch}(x)\)
Inverse Trigonometric Functions
\(\displaystyle \int \arcsin(x) \, dx = x\cdot\arcsin(x)+\sqrt{1-x^2}\)
\(\displaystyle \int \arctan(x) \, dx = x\cdot\arctan(x)-\frac12 \ln(1+x^2)\)
Rational Functions
\(\displaystyle \int \frac{1}{a^2+x^2} \, dx = \frac{1}{a}\arctan\left(\frac{x}{a}\right)\)
\(\displaystyle \int \frac{1}{a^2-x^2} \, dx = \frac{1}{2a}\ln\left(\left|\frac{x+a}{x-a}\right|\right) = \frac{1}{a}\mbox{argtanh}\left(\frac{x}{a}\right)\)
\(\displaystyle \int \frac{1}{(a^2+x^2)^2} \, dx = \frac{1}{2a^3}\left[\frac{ax}{a^2+x^2}+\arctan\left(\frac{x}{a}\right)\right]\)
\(\displaystyle \int \frac{1}{(x-a)(x-b)} \, dx = \frac{1}{a-b}\ln\left(\left|\frac{x-a}{x-b}\right|\right)\)
Radical Functions
\(\displaystyle \int \sqrt{x^2\pm a^2} \, dx = \frac{x}{2}\sqrt{x^2\pm a^2}+\frac{a^2}{2}\ln\left(x+\sqrt{x^2\pm a^2}\right)\)
\(\displaystyle \int \sqrt{a^2-x^2} \, dx = \frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\arctan\left(\frac{x}{\sqrt{a^2-x^2}}\right)\)
\(\displaystyle \int \frac{1}{\sqrt{x^2\pm a^2}} \, dx = \ln\left(x+\sqrt{x^2\pm a^2}\right)\)
\(\displaystyle \int \frac{1}{\sqrt{a^2-x^2}} \, dx = \arcsin\left(\frac{x}{a}\right)\)
\(\displaystyle \int \frac{1}{\sqrt{x^2 + a^2}} \, dx = \mbox{argsinh}\left(\frac{x}{a}\right)\)
\(\displaystyle \int \frac{1}{x \sqrt{x^2-a^2}} \, dx = \frac{1}{a}\mbox{arcsec}\left(\frac{x}{a}\right)\) \quad (\)x > a\))
Products of Exponentials and Trigonometric or Hyperbolic Functions
\(\displaystyle \int e^{ax}\sin(bx) \, dx = \frac{e^{ax}}{a^2+b^2}\left[a\sin(bx)-b\cos(bx)\right]\)
\(\displaystyle \int e^{ax}\cos(bx) \, dx = \frac{e^{ax}}{a^2+b^2}\left[a\cos(bx)+b\sin(bx)\right]\)
\(\displaystyle \int e^{ax}\sinh(bx) \, dx = \frac{e^{ax}}{a^2-b^2}\left[a\sinh(bx)-b\cosh(bx)\right]\)
\(\displaystyle \int e^{ax}\cosh(bx) \, dx = \frac{e^{ax}}{a^2-b^2}\left[a\cosh(bx)-b\sinh(bx)\right]\)