Skip to main content\(\require{cancel}\newcommand{\highlight}[1]{{\color{blue}{#1}}}
\newcommand{\apex}{A\kern -1pt \lower -2pt\mbox{P}\kern -4pt \lower .7ex\mbox{E}\kern -1pt X}
\newcommand{\colorlinecolor}{blue!95!black!30}
\newcommand{\bwlinecolor}{black!30}
\newcommand{\thelinecolor}{\colorlinecolor}
\newcommand{\colornamesuffix}{}
\newcommand{\linestyle}{[thick, \thelinecolor]}
\newcommand{\bmx}[1]{\left[\hskip -3pt\begin{array}{#1} }
\newcommand{\emx}{\end{array}\hskip -3pt\right]}
\newcommand{\ds}{\displaystyle}
\newcommand{\fp}{f'}
\newcommand{\fpp}{f''}
\newcommand{\lz}[2]{\frac{d#1}{d#2}}
\newcommand{\lzn}[3]{\frac{d^{#1}#2}{d#3^{#1}}}
\newcommand{\lzo}[1]{\frac{d}{d#1}}
\newcommand{\lzoo}[2]{{\frac{d}{d#1}}{\left(#2\right)}}
\newcommand{\lzon}[2]{\frac{d^{#1}}{d#2^{#1}}}
\newcommand{\lzoa}[3]{\left.{\frac{d#1}{d#2}}\right|_{#3}}
\newcommand{\plz}[2]{\frac{\partial#1}{\partial#2}}
\newcommand{\plzoa}[3]{\left.{\frac{\partial#1}{\partial#2}}\right|_{#3}}
\newcommand{\inflim}[1][n]{\lim\limits_{#1 \to \infty}}
\newcommand{\infser}[1][1]{\sum_{n=#1}^\infty}
\newcommand{\primeskip}{\hskip.75pt}
\newcommand{\Fp}{F\primeskip'}
\newcommand{\Fpp}{F\primeskip''}
\newcommand{\yp}{y\primeskip'}
\newcommand{\gp}{g\primeskip'}
\newcommand{\dx}{\Delta x}
\newcommand{\dy}{\Delta y}
\newcommand{\ddz}{\Delta z}
\newcommand{\thet}{\theta}
\newcommand{\norm}[1]{\left\lVert#1\right\rVert}
\newcommand{\vnorm}[1]{\left\lVert\vec #1\right\rVert}
\newcommand{\snorm}[1]{\left|\left|\,#1\,\right|\right|}
\newcommand{\la}{\left\langle}
\newcommand{\ra}{\right\rangle}
\newcommand{\dotp}[2]{\vec #1 \cdot \vec #2}
\newcommand{\proj}[2]{\text{proj}_{\,\vec #2}{\,\vec #1}}
\newcommand{\crossp}[2]{\vec #1 \times \vec #2}
\newcommand{\veci}{\hat\imath}
\newcommand{\vecj}{\hat\jmath}
\newcommand{\veck}{\hat{k}}
\newcommand{\vecu}{\vec u}
\newcommand{\vecv}{\vec v}
\newcommand{\vecw}{\vec w}
\newcommand{\vecx}{\vec x}
\newcommand{\vecy}{\vec y}
\newcommand{\vrp}{\vec r\primeskip '}
\newcommand{\vrpp}{\vec r\primeskip ''}
\newcommand{\vsp}{\vec s\primeskip '}
\newcommand{\vrt}{\vec r(t)}
\newcommand{\vst}{\vec s(t)}
\newcommand{\vvt}{\vec v(t)}
\newcommand{\vat}{\vec a(t)}
\newcommand{\px}{\partial x}
\newcommand{\py}{\partial y}
\newcommand{\pz}{\partial z}
\newcommand{\pf}{\partial f}
\newcommand{\unittangent}{\vec{T}}
\newcommand{\unitnormal}{\vec{N}}
\newcommand{\unittangentprime}{\vec{T}\primeskip '}
\newcommand{\mathN}{\mathbb{N}}
\newcommand{\surfaceS}{\mathcal{S}}
\newcommand{\zerooverzero}{\ds \raisebox{8pt}{\text{``\ }}\frac{0}{0}\raisebox{8pt}{\textit{ ''}}}
\newcommand{\deriv}[2]{\myds\frac{d}{dx}\left(#1\right)=#2}
\newcommand{\myint}[2]{\myds\int #1\, dx= {\ds #2}}
\newcommand{\abs}[1]{\left\lvert #1\right\rvert}
\newcommand{\sech}{\operatorname{sech}}
\newcommand{\csch}{\operatorname{csch}}
\newcommand{\curl}{\operatorname{curl}}
\newcommand{\divv}{\operatorname{div}}
\newcommand{\Hess}{\operatorname{Hess}}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Subsection A.6 Integration Rules
Subsubsection A.6.1 Basic Rules
\(\displaystyle \int c\cdot f(x)\ dx=\) \(\displaystyle c\int f(x)\ dx\)
\(\displaystyle \int f(x)\pm g(x)\ dx=\) \(\displaystyle \) \(\displaystyle \int f(x)\ dx \pm \int g(x)\ dx\)
\(\displaystyle \int 0\ dx=\) \(\displaystyle C\)
\(\displaystyle \int 1\ dx=\) \(\displaystyle x+C\)
\(\displaystyle \int x^n\ dx=\) \(\displaystyle \frac{1}{n+1}x^{n+1}+C, \ n\ne -1\)
\(\displaystyle \int e^x\ dx=\) \(\displaystyle e^x+C\)
\(\displaystyle \int a^x\ dx=\) \(\displaystyle \frac{1}{\ln a}\cdot a^x+C\)
\(\displaystyle \int \frac{1}{x}\ dx=\) \(\displaystyle \ln |x| + C\)
Subsubsection A.6.2 Trigonometric Functions
\(\displaystyle \int \cos x\ dx=\) \(\displaystyle \sin x+C\)
\(\displaystyle \int \sin x\ dx=\) \(\displaystyle -\cos x+C\)
\(\displaystyle \int \tan x\ dx=\) \(\displaystyle -\ln |\cos x|+C\)
\(\displaystyle \int \sec x\ dx=\) \(\displaystyle \ln |\sec x+\tan x|+C\)
\(\displaystyle \int \csc x\ dx=\) \(\displaystyle -\ln |\csc x+\cot x|+C\)
\(\displaystyle \int \cot x\ dx=\) \(\displaystyle \ln |\sin x|+C\)
\(\displaystyle \int \sec ^2 x\ dx=\) \(\displaystyle \tan x+C\)
\(\displaystyle \int \csc ^2x\ dx=\) \(\displaystyle -\cot x+C\)
\(\displaystyle \int \sec x\tan x\ dx=\) \(\displaystyle \sec x+C\)
\(\displaystyle \int \csc x\cot x\ dx=\) \(\displaystyle -\csc x+C\)
\(\displaystyle \int \cos ^2x\ dx=\) \(\displaystyle \frac{1}{2}x+\frac{1}{4}\sin \big (2x\big )+C\)
\(\displaystyle \int \sin ^2x\ dx=\) \(\displaystyle \frac{1}{2}x-\frac{1}{4}\sin \big (2x\big )+C\)
Subsubsection A.6.3 Common Trig Substitutions
\(\displaystyle \int \frac{1}{x^2+a^2}\ dx=\) \(\displaystyle \frac{1}{a}\tan ^{-1}\left(\frac{x}{a}\right)+C\)
\(\displaystyle \int \frac{1}{\sqrt{a^2-x^2}}\ dx=\) \(\displaystyle \sin ^{-1}\left(\frac{x}{a}\right)+C\)
\(\displaystyle \int \frac{1}{x\sqrt{x^2-a^2}}\ dx=\) \(\displaystyle \frac{1}{a}\sec ^{-1}\left(\frac{x}{a}\right)+C\)
\(\displaystyle \int \frac{1}{\sqrt{x^2-a^2}}\ dx=\) \(\displaystyle \ln \big |x+\sqrt{x^2-a^2}\big |+C\)
\(\displaystyle \int \frac{1}{\sqrt{x^2+a^2}}\ dx=\) \(\displaystyle \ln \big |x+\sqrt{x^2+a^2}\big |+C\)
\(\displaystyle \int \frac{1}{a^2-x^2}\ dx=\) \(\displaystyle \frac{1}{2}\ln \left|\frac{a+x}{a-x}\right|+C\)
\(\displaystyle \int \frac{1}{x\sqrt{a^2-x^2}}\ dx=\) \(\displaystyle \frac{1}{a}\ln \left(\frac{x}{a+\sqrt{a^2-x^2}}\right)+C\)
\(\displaystyle \int \frac{1}{x\sqrt{x^2+a^2}}\ dx=\) \(\displaystyle \frac{1}{a}\ln \left|\frac{x}{a+\sqrt{x^2+a^2}}\right|+C\)
Subsubsection A.6.4 Hyperbolic Trigonometric Functions
\(\displaystyle \int \cosh x\ dx=\) \(\displaystyle \sinh x+C\)
\(\displaystyle \int \sinh x\ dx=\) \(\displaystyle \cosh x+C\)
\(\displaystyle \int \tanh x\ dx=\) \(\displaystyle \ln (\cosh x)+C\)
\(\displaystyle \int \coth x\ dx=\) \(\displaystyle \ln |\sinh x|+C\)
