From 621e67426beeefa6e5bc02f220ab6d41cb17693a Mon Sep 17 00:00:00 2001 From: juripfammatter Date: Sat, 20 Jan 2024 15:31:44 +0100 Subject: [PATCH] removed duplicates and added a few integrals over special intervals --- src/sections/Appendix.tex | 16 +++++++++------- 1 file changed, 9 insertions(+), 7 deletions(-) diff --git a/src/sections/Appendix.tex b/src/sections/Appendix.tex index a9f233f..513b3e8 100644 --- a/src/sections/Appendix.tex +++ b/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}