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Table of Integrals II - Trig WO.tex
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\documentclass[10pt]{article} \usepackage{amssymb,amsmath} \usepackage[hmargin=1cm,vmargin=0.5cm]{geometry} \begin{document} \noindent Table of Integrals \--- Trigonometric Functions \--- The Working Out\\ Note: $t=\tan\frac{ax}{2}$, $\sin ax=\frac{2t}{1+t^2}$, $\cos ax=\frac{1-t^2}{1+t^2}$, $\tan ax=\frac{2t}{1-t^2}$, $dx=\frac{\:1\:}{a}\cdot\frac{2}{1+t^2}\:dt$ \begin{align*} &\int\sin ax\:dx=-\frac{\:1\:}{a}\cos ax+C\\ \\ &\int\cos ax\:dx=\frac{\:1\:}{a}\sin ax+C\\ \\ &\int\tan ax\:dx=\int\frac{\sin ax}{\cos ax}\:dx=\int\frac{-\frac{\:1\:}{a}(\cos ax)'}{\cos ax}\:dx=-\frac{\:1\:}{a}\ln\left|\cos ax\right|+C\:\:\Big(=\frac{\:1\:}{a}\ln\left|\sec ax\right|+C\Big)\\ \\ &\int\cot ax\:dx=\int\frac{\cos ax}{\sin ax}\:dx=\int\frac{\frac{\:1\:}{a}(\sin ax)'}{\sin ax}\:dx=\frac{\:1\:}{a}\ln\left|\sin ax\right|+C\\ \\ &\int\sec ax\:dx=\int\sec ax\cdot\frac{\sec ax + \tan ax}{\sec ax + \tan ax}\:dx=\int\frac{\sec^2 ax+\sec ax\tan ax}{\sec ax + \tan ax}\:dx\\ &\quad=\int\frac{\frac{\:1\:}{a}(\tan ax + \sec ax)'}{\sec ax + \tan ax}\:dx=\frac{\:1\:}{a}\ln\left|\sec ax+\tan ax\right|+C\\ \\ &\int\sec ax\:dx=\int\frac{\rlap{--------}1+t^2}{1-t^2}\cdot\frac{\:1\:}{a}\cdot\frac{2}{\rlap{--------}1+t^2}\:dt=\frac{\:1\:}{a}\int\left(\frac{1}{1+t}+\frac{1}{1-t}\right)\:dt\\ &\quad=\frac{\:1\:}{a}\Big(\ln|1+t|-\ln|1-t|\Big)+C=\frac{\:1\:}{a}\ln\left|\frac{1+t}{1-t}\right|+C=\frac{\:1\:}{a}\ln\left|\frac{1+\tan\frac{ax}{2}}{1-\tan\frac{ax}{2}}\right|+C\\ \\ &\int\csc ax\:dx=\int\csc ax\cdot\frac{\csc ax - \cot ax}{\csc ax - \cot ax}\:dx=\int\frac{\csc^2 ax - \csc ax\cot ax}{\csc ax - \cot ax}\:dx\\ &\quad=\int\frac{\frac{\:1\:}{a}(\cot ax - \csc ax)'}{\csc ax - \cot ax}\:dx=\frac{\:1\:}{a}\ln\left|\csc ax-\cot ax\right|+C\:\: \Big(=-\frac{\:1\:}{a}\ln\left|\csc ax+\cot ax\right|+C\Big)\\ \\ &\int\csc ax\:dx=\int\frac{\rlap{--------}1+t^2}{\not{2}t}\cdot\frac{\:1\:}{a}\cdot\frac{\not{2}}{\rlap{--------}1+t^2}\:dt=\frac{\:1\:}{a}\int\frac{\:1\:}{t}\:dt =\frac{\:1\:}{a}\ln|t|+C=\frac{\:1\:}{a}\ln\left|\tan\frac{ax}{2}\right|+C \end{align*} \begin{align*} &I_n=\int\sin^n ax\:dx\quad\Rightarrow\quad I_n=\tfrac{n-1}{n}I_{n-2}-\tfrac{\:1\:}{na}\sin^{n-1}ax\cdot\cos ax\\ &I_n=\int\sin^{n-1}ax\cdot\sin ax\:dx\\ &\quad=\sin^{n-1}ax\cdot\left(-\frac{\:1\:}{a}\cos ax\right)-\int\left(-\frac{\:1\:}{\not{a}}\cos ax\right)\cdot(n-1)\sin^{n-2}ax\cdot\cos ax\cdot \not{a}\:dx\\ &\quad=-\frac{\:1\:}{a}\sin^{n-1} ax\cdot\cos ax+(n-1)\int\sin^{n-2}ax\cdot\cos^2 ax\:dx\\ &\quad=-\frac{\:1\:}{a}\sin^{n-1} ax\cdot\cos ax+(n-1)\int\sin^{n-2}ax\cdot(1-\sin^2 ax)\:dx\\ &\quad=-\frac{\:1\:}{a}\sin^{n-1} ax\cdot\cos ax+(n-1)\int\sin^{n-2}ax\:dx-(n-1)\int\sin^n ax\:dx\\ &\quad=-\frac{\:1\:}{a}\sin^{n-1} ax\cdot\cos ax+(n-1)I_{n-2}-(n-1)I_n\\ &nI_n=-\frac{\:1\:}{a}\sin^{n-1} ax\cdot\cos ax+(n-1)I_{n-2}\\ &I_n=\tfrac{n-1}{n}I_{n-2}-\tfrac{\:1\:}{na}\sin^{n-1}ax\cdot\cos ax \end{align*} \begin{align*} &I_n=\int\cos^n ax\:dx\quad\Rightarrow\quad I_n=\tfrac{n-1}{n}I_{n-2}+\tfrac{\:1\:}{na}\cos^{n-1}ax\cdot\sin ax\\ &I_n=\int\cos^{n-1}ax\cdot\cos ax\:dx\\ &\quad=\cos^{n-1}ax\cdot\left(\frac{\:1\:}{a}\sin ax\right)-\int\left(\frac{\:1\:}{\not{a}}\sin ax\right)\cdot(n-1)\cos^{n-2}ax\cdot(-\sin ax)\cdot \not{a}\:dx\\ &\quad=\frac{\:1\:}{a}\cos^{n-1} ax\cdot\sin ax+(n-1)\int\cos^{n-2}ax\cdot\sin^2 ax\:dx\\ &\quad=\frac{\:1\:}{a}\cos^{n-1} ax\cdot\sin ax+(n-1)\int\cos^{n-2}ax\cdot(1-\cos^2 ax)\:dx\\ &\quad=\frac{\:1\:}{a}\cos^{n-1} ax\cdot\sin ax+(n-1)\int\cos^{n-2}ax\:dx-(n-1)\int\cos^n ax\:dx\\ &\quad=\frac{\:1\:}{a}\cos^{n-1} ax\cdot\sin ax+(n-1)I_{n-2}-(n-1)I_n\\ &nI_n=\frac{\:1\:}{a}\cos^{n-1} ax\cdot\sin ax+(n-1)I_{n-2}\\ &I_n=\tfrac{n-1}{n}I_{n-2}+\tfrac{\:1\:}{na}\cos^{n-1}ax\cdot\sin ax \end{align*} \begin{align*} &I_n=\int\tan^n ax\:dx\quad\Rightarrow\quad I_n=\tfrac{1}{(n-1)a}\tan^{n-1}ax-I_{n-2}\\ &I_n=\int\left(\frac{\tan^{n-1}ax}{\sec ax}\right)\cdot\sec ax\:\tan ax\:dx\\ &\quad=\left(\frac{\tan^{n-1}ax}{\sec ax}\right)\cdot\frac{\:1\:}{a}\sec ax-\int\frac{\:1\:}{\not{a}}\sec ax\cdot\frac{1}{\sec^2 ax}\left(\sec ax\cdot(n-1)\tan^{n-2}ax\:\sec^2 ax\cdot \not{a}-\tan^{n-1}ax\cdot\sec ax\:\tan ax\cdot \not{a}\right)\:dx\\ &\quad=\frac{\:1\:}{a}\tan^{n-1}ax-\int\left[(n-1)\tan^{n-2}ax\:\sec^2 ax-\tan^n ax\right]\:dx\\ &\quad=\frac{\:1\:}{a}\tan^{n-1}ax-\int(n-1)\tan^{n-2}ax\cdot(\tan^2 ax+1)+\int\tan^n ax\:dx\\ &\quad=\frac{\:1\:}{a}\tan^{n-1}ax-\int(n-1)\tan^n ax\:dx-\int(n-1)\tan^{n-2} ax\:dx+\int\tan^n ax\:dx\\ &\quad=\frac{\:1\:}{a}\tan^{n-1}ax-(n-1)I_n-(n-1)I_{n-2}+I_n\\ &(n-1)I_n=\frac{\:1\:}{a}\tan^{n-1}ax-(n-1)I_{n-2}\\ &I_n=\tfrac{1}{(n-1)a}\tan^{n-1}ax-I_{n-2} \end{align*} \begin{align*} &I_n=\int\sec^n ax\:dx\quad\Rightarrow\quad I_n=\tfrac{1}{(n-1)a}\sec^{n-2}ax\cdot\tan ax+\tfrac{n-2}{n-1}I_{n-2}\\ &I_n=\int\sec^{n-2}ax\cdot\sec^2 ax\:dx\\ &\quad=\sec^{n-2}ax\cdot\frac{\:1\:}{a}\tan ax-\int\frac{\:1\:}{\not{a}}\tan ax\cdot(n-2)\sec^{n-3}ax\cdot\sec ax\:\tan ax\cdot \not{a}\:dx\\ &\quad=\frac{\:1\:}{a}\sec^{n-2}ax\:\tan ax-(n-2)\int\sec^{n-2}ax\cdot\tan^2 ax\:dx\\ &\quad=\frac{\:1\:}{a}\sec^{n-2}ax\:\tan ax-(n-2)\int\sec^{n-2}ax\cdot(\sec^2 ax-1)\:dx\\ &\quad=\frac{\:1\:}{a}\sec^{n-2}ax\:\tan ax-(n-2)\int\sec^n ax\:dx+(n-2)\int\sec^{n-2}ax\:dx\\ &\quad=\frac{\:1\:}{a}\sec^{n-2}ax\:\tan ax-(n-2)I_n+(n-2)I_{n-2}\\ &(n-1)I_n=\frac{\:1\:}{a}\sec^{n-2}ax\:\tan ax+(n-2)I_{n-2}\\ &I_n=\tfrac{1}{(n-1)a}\sec^{n-2}ax\cdot\tan ax+\tfrac{n-2}{n-1}I_{n-2} \end{align*} \end{document}