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Table of Integrals II - Trig.tex
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List of integrals of trigonometric functions
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\documentclass[12pt]{article} \usepackage{amssymb,amsmath} \usepackage[hmargin=1cm,vmargin=1cm]{geometry} \begin{document} Table of Integrals - Trigonometric Functions\\ (Note: `+ C' is omitted.) \begin{align*} &\int\sin ax\:dx=-\frac{\:1\:}{a}\cos ax\\ &\int\cos ax\:dx=\frac{\:1\:}{a}\sin ax\\ &\int\tan ax\:dx=\int\frac{\sin ax}{\cos ax}\:dx=-\frac{\:1\:}{a}\ln\left|\cos ax\right|\:\:\Big(=\frac{\:1\:}{a}\ln\left|\sec ax\right|\Big)\\ &\int\cot ax\:dx=\int\frac{\cos ax}{\sin ax}\:dx=\frac{\:1\:}{a}\ln\left|\sin ax\right|\\ &\int\sec ax\:dx=\frac{\:1\:}{a}\ln\left|\sec ax+\tan ax\right|\\ &\int\sec ax\:dx=\frac{\:1\:}{a}\ln\left|\frac{1+\tan\frac{ax}{2}}{1-\tan\frac{ax}{2}}\right|\\ &\int\csc ax\:dx=\frac{\:1\:}{a}\ln\left|\csc ax-\cot ax\right|\:\: \Big(=-\frac{\:1\:}{a}\ln\left|\csc ax+\cot ax\right|\Big)\\ &\int\csc ax\:dx=\frac{\:1\:}{a}\ln\left|\tan\frac{ax}{2}\right|\\ \\ &\int\sin^2 ax\:dx=\int\frac{\:1\:}{2}\left(1-\cos 2ax\right)\:dx=\frac{\:1\:}{2}\left(x-\frac{\:1\:}{2a}\sin 2ax\right)\\ &\int\cos^2 ax\:dx=\int\frac{\:1\:}{2}\left(1+\cos 2ax\right)\:dx=\frac{\:1\:}{2}\left(x+\frac{\:1\:}{2a}\sin 2ax\right)\\ &\int\tan^2 ax\:dx=\int\left(\sec^2 ax-1\right)\:dx=\frac{\:1\:}{a}\tan ax-x\\ &\int\cot^2 ax\:dx=\int\left(\csc^2 ax-1\right)\:dx=-\frac{\:1\:}{a}\cot ax-x\\ &\int\sec^2 ax\:dx=\frac{\:1\:}{a}\tan ax\\ &\int\csc^2 ax\:dx=-\frac{\:1\:}{a}\cot ax\\ \\ &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\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\tan^n ax\:dx\quad\Rightarrow\quad I_n=\tfrac{1}{(n-1)a}\tan^{n-1}ax-I_{n-2}\\ &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}\\ \end{align*} \end{document}