Merge pull request #72 from MeierTobias/appendix/integrals

removed duplicates and added a few integrals over special intervals

src/sections/Appendix.tex

@@ -275,17 +275,15 @@ \subsubsection{Integration of trig.\ functions on special intervals}
 \begin{align*}
  & \int_0^{\infty} \frac{\sin(ax)}{x} dx & & =\frac{\pi}{2};\quad a>0 \\
  & \int_0^{\infty} \sin(x^2) dx & & =\int_0^{\infty}\cos(x^2)dx=\frac{1}{2}\sqrt{\frac{\pi}{2}} \\
- & \int_0^{\infty} e^{-ax}x^n dx & & =\frac{n!}{a^{n+1}},\quad a>0 \\
- & \int_0^{\infty} e^{-ax^2} dx & & =\frac{1}{2}\sqrt{\frac{\pi}{a}},\quad a>0 \\
  & \int_{-\pi}^{\pi} e^{ijx} \text{d}x & & = \begin{cases}2\pi & \text{if } j=0\\ 0 & \text{if } j\neq 0\end{cases}
 \end{align*}
 
 
 \renewcommand{\arraystretch}{1.5}
-\setlength\tabcolsep{10pt} % default value: 6pt
+\setlength\tabcolsep{8pt} % default value: 6pt
 \begin{tabularx}{\linewidth}{@{}lccccccc@{}}
  & $\int\limits_0^{\frac{\pi}{4}} $ & $\int\limits_0^{\frac{\pi}{2}}$ & $\int\limits_0^{\pi}$ & $\int\limits_0^{2\pi}$ & $\int\limits_{-\frac{\pi}{4}}^{\frac{\pi}{4}} $ & $\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}} $ & $\int\limits_{-\pi}^{\pi}$ \\ 
- \cmidrule{2-7}
+ \cmidrule{2-8}
  $\sin$ & $\frac{\sqrt{2}-1}{\sqrt{2}}$ & 1 & 2 & 0 & 0 & 0 & 0 \\ 
  $\sin^2$ & $\frac{\pi-2}{8}$ & $\frac{\pi}{4}$ & $\frac{\pi}{2}$ & $\pi$ & $\frac{\pi-2}{4}$ & $\frac{\pi}{2}$ & $\pi$ \\
  $\sin^3$ & $\frac{8-5\sqrt{2}}{12}$ & $\frac{2}{3}$ & $\frac{4}{3}$ & 0 & 0 & 0 & 0 \\
@@ -311,8 +309,6 @@ \subsubsection{Intergation of power, rational, exponential and logarithmic funct
  & & & \qquad +\frac{b^2{(ax+b)}^{n+1}}{(n+1)a^3}+C \\
  & \int \frac{1}{ax+b}dx & & =\frac{1}{a}\ln\vert ax+b \vert +C \\
  & \int \frac{ax+b}{cx+d}dx & & =\frac{ax}{c}-\frac{ad-bc}{c^2}\ln\vert cx+d \vert +C \\
- & \int \frac{1}{x^2+a^2}dx & & =\frac{1}{a}\arctan\left(\frac{x}{a}\right) +C \\
- & \int \frac{1}{x^2-a^2}dx & & =\frac{1}{2a}\ln\big\vert \frac{x-a}{x+a}\big\vert +C \\
  & \int \frac{x}{{(ax+b)}^n}dx & & =-\frac{1}{(n-2)a^2{(ax+b)}^{n-2}} \dots \\
  & & & \qquad +\frac{b}{(n-1)a^2{(ax+b)}^{n-1}}+C \\
  & \int \frac{x}{x^2+a}dx & & =\frac{1}{2}\ln\vert x^2+a \vert+C \\
@@ -363,7 +359,12 @@ \subsubsection{Intergation of power, rational, exponential and logarithmic funct
 
 \subsubsection{Integration over special intervals}
 \begin{align*}
- & \int_{-\infty}^{\infty}\frac{1}{1+x^2}dx & & =\pi
+ & \int_{-\infty}^{\infty}\frac{1}{1+x^2}dx & & = \pi \\
+ & \int_0^{\infty} e^{-ax}x^n dx & & =\frac{n!}{a^{n+1}},\quad a>0 \\
+ & \int_0^{\infty} e^{-ax^2} dx & & =\frac{1}{2}\sqrt{\frac{\pi}{a}},\quad a>0 \\
+ & \int_{-\infty}^{\infty} e^{-ax^2} dx & & =\sqrt{\frac{\pi}{a}},\quad a>0 \\
+ & \int_{-\infty}^{\infty}e^{-ax^2}e^{-iwx} dx & & = \sqrt{\frac{\pi}{a}}e^{\frac{w^2}{4a}} \\
+ & \int_{-\infty}^{\infty}e^{-(ax^2+bx+c)}dx & & = \sqrt{\frac{\pi}{a}}e^{\frac{b^2}{4a}-c} \\
 \end{align*}
 
 \subsection{Various functions and their properties}
@@ -428,6 +429,7 @@ \subsubsection{Complex numbers}
 
 
 
+
 \end{tabular}
 
 \textbf{Operations}