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    pdftitle={Integral Table from http://integral-table.com},    % title
    pdfauthor={Shapiro},     % author
    pdfsubject={Table of Integrals},   % subject of the document
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    pdfkeywords={CSUN, Integrals, Table of Integrals, Math 280, Math 351, Differential Equations}, % list of keywords
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\date{} % delete this line to display the current date                                                                                             

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\begin{document}


\begin{center}
\LARGE
Table of Integrals\footnote{Most formulas are from  \url{http://integral-table.com}}



\end{center}
\normalsize
\begin{multicols}{2}

%%\begin{footnotesize}

\begin{equation}
log_a(x) = \frac{ln x}{ln a}
\end{equation}

\begin{center} \textbf{Basic Forms}
\end{center}


\begin{equation}
\int u dv = uv - \int v du
\end{equation}



\begin{center} \textbf{Integrals with Roots} \end{center}



\begin{equation}
\int x\sqrt{x-a} dx = \frac{2}{3}a(x-a)^{3/2}+\frac{2}{5}(x-a)^{5/2}
\end{equation}


\begin{align}
\int \sqrt{\frac{x}{a-x}}dx &=  -\sqrt{x(a-x)}
%\nonumber \\&
-a\tan^{-1}\frac{\sqrt{x(a-x)}}{x-a}
\end{align}

\begin{align}
\int \sqrt{\frac{x}{a+x}}dx &=  \sqrt{x(a+x)} 
%\nonumber \\& 
-a\ln \left [ \sqrt{x} + \sqrt{x+a}\right] 
\end{align}

\begin{align}
\int &x \sqrt{ax + b}dx =
%\nonumber \\& 
\frac{2}{15 a^2}(-2b^2+abx + 3 a^2 x^2)
\sqrt{ax+b}
\end{align}

\begin{align}
\int \sqrt{x(ax+b)} dx &= \frac{1}{4a^{3/2}}\left[(2ax + b)\sqrt{ax(ax+b)} \right. \nonumber
\\& \left.
-b^2 \ln \left| a\sqrt{x} + \sqrt{a(ax+b)} \right| \right ] 
\end{align}



\begin{align}
\int\sqrt{x^2 \pm a^2} dx &= \frac{1}{2}x\sqrt{x^2\pm a^2} 
%\nonumber \\&
\pm\frac{1}{2}a^2 \ln \left | x + \sqrt{x^2\pm a^2} \right | 
\end{align}

\begin{align}
\int  \sqrt{a^2 - x^2} dx &= \frac{1}{2} x \sqrt{a^2-x^2} 
%\nonumber \\ &
+\frac{1}{2}a^2\tan^{-1}\frac{x}{\sqrt{a^2-x^2}}
\end{align}




\begin{equation}
\int \frac{1}{\sqrt{x^2 \pm a^2}} dx = \ln \left | x + \sqrt{x^2 \pm a^2} \right | 
\end{equation}

\begin{equation}
\int \frac{1}{\sqrt{a^2 - x^2}} dx = \sin^{-1}\frac{x}{a} 
\end{equation}


\vfill


\begin{center} \textbf{Integrals with Logarithms} \end{center}

\begin{equation}
\int \ln x dx = x \ln x - x 
\end{equation}



\begin{equation}
\int \ln (ax + b) dx = \left ( x + \frac{b}{a} \right) \ln (ax+b) - x , a\ne 0
\end{equation}

\begin{align}
\int \ln  ( x^2 + a^2 )\hspace{.5ex}\text{dx} = x \ln (x^2 + a^2  ) +2a\tan^{-1} \frac{x}{a} - 2x 
\end{align}

\begin{align}
\int \ln  ( x^2 - a^2 )\hspace{.5ex}\text{dx} = x \ln (x^2 - a^2  ) +a\ln \frac{x+a}{x-a} - 2x \end{align}



\begin{align}
\int x \ln (ax + b) dx &= \frac{bx}{2a}-\frac{1}{4}x^2 \nonumber
\\&
+\frac{1}{2}\left(x^2-\frac{b^2}{a^2}\right)\ln (ax+b) 
\end{align}


\columnbreak


 
\begin{center} \textbf{Integrals with Exponentials} \end{center}

\begin{equation}
\int a^{x} dx = \frac{a^x}{ln a},~ a > 0, a \not= 1 
\end{equation}

\begin{align}
\int \sqrt{x} e^{ax} dx &= \frac{1}{a}\sqrt{x}e^{ax} 
+\frac{i\sqrt{\pi}}{2a^{3/2}}
\text{erf}\left(i\sqrt{ax}\right), \nonumber \\&
\text{ where erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt
\end{align}

\begin{equation}
\int x e^x dx = (x-1) e^x 
\end{equation}

\begin{equation}
\int x e^{ax} dx = \left(\frac{x}{a}-\frac{1}{a^2}\right) e^{ax} 
\end{equation}

\begin{equation}
\int x^2 e^{x} dx = \left(x^2 - 2x + 2\right) e^{x}
\end{equation}

\begin{equation}
\int x^2 e^{ax} dx = \left(\frac{x^2}{a}-\frac{2x}{a^2}+\frac{2}{a^3}\right) e^{ax} 
\end{equation}

\begin{equation}
\int x^3 e^{x} dx = \left(x^3-3x^2 + 6x - 6\right) e^{x} 
\end{equation}
 
\begin{equation}
\int x^n e^{ax}\hspace{1pt}\text{d}x = \dfrac{x^n e^{ax}}{a} - 
\dfrac{n}{a}\int x^{n-1}e^{ax}\hspace{1pt}\text{d}x
\end{equation} 
 
\begin{equation}
\begin{split}
\int x^n e^{ax}\hspace{2pt}\text{d}x = \frac{(-1)^n}{a^{n+1}}\Gamma[1+n,-ax], \\
 \text{ where } \Gamma(a,x)=\int_x^{\infty} t^{a-1}e^{-t}\hspace{2pt}\text{d}t
 \end{split}
 \end{equation}

\begin{equation}
\int e^{ax^2}\hspace{1pt}\text{d}x = -\frac{i\sqrt{\pi}}{2\sqrt{a}}\text{erf}\left(ix\sqrt{a}\right) 
\end{equation}

\begin{equation}
\int e^{-ax^2}\hspace{1pt}\text{d}x = \frac{\sqrt{\pi}}{2\sqrt{a}}\text{erf}\left(x\sqrt{a}\right) 
\end{equation}



\begin{equation}
\int x^2 e^{-ax^2}\ \text{dx} = \dfrac{1}{4}\sqrt{\dfrac{\pi}{a^3}}\text{erf}(x\sqrt{a}) -\dfrac{x}{2a}e^{-ax^2}
\end{equation}


\begin{center} \textbf {Integrals with Trigonometric Functions} \end{center}




\begin{equation}
\int \sec^2 x dx =  \tan x 
\end{equation}


\begin{equation}
\int \tan x dx = - \ln \cos x 
\end{equation}




 
\begin{center} \textbf{Products of Trigonometric Functions and Exponentials} \end{center}


\begin{equation}
\int e^{bx} \cos ax dx = \frac{1}{a^2 + b^2} e^{bx} ( a \sin ax + b \cos ax ) 
\end{equation}

\begin{equation}
\int x e^x \sin x dx = \frac{1}{2}e^x (\cos x - x \cos x + x \sin x) 
\end{equation}

\begin{equation}
\int x e^x \cos x dx = \frac{1}{2}e^x (x \cos x 
- \sin x + x \sin x) 
\end{equation}


\begin{center} \textbf{Definite integral lacking closed-form antiderivatives} \end{center}


\begin{equation}
\int_{\infty}^\infty e^{-x^2} = \sqrt{\pi}
\end{equation}



%%\end{footnotesize}
\end{multicols}
\end{document}  