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A.4: Tabla de Integrales

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A lo largo de esta tabla,$$a$$ y$$b$$ se les dan constantes, independientes$$x$$ y$$C$$ es una constante arbitraria.

 $$f(x)$$ $$F(x)=\int f(x)\ \mathrm{d}{x}$$ $$af(x)+bg(x)$$ $$a\int f(x)\ \mathrm{d}{x} +b\int g(x)\ \mathrm{d}{x} \ +\ C$$ $$f(x)+g(x)$$ $$\int f(x)\ \mathrm{d}{x} +\int g(x)\ \mathrm{d}{x} \ +\ C$$ $$f(x)-g(x)$$ $$\int f(x)\ \mathrm{d}{x} -\int g(x)\ \mathrm{d}{x} \ +\ C$$ $$af(x)$$ $$a\int f(x)\ \mathrm{d}{x} \ +\ C$$ $$u(x)v'(x)$$ $$u(x)v(x)-\int u'(x)v(x)\ \mathrm{d}{x} \ +\ C$$ $$f\big(y(x)\big)y'(x)$$ $$F\big(y(x)\big)\hbox{ where }F(y)=\int f(y)\ \mathrm{d}{y}$$ $$a$$ $$ax+C$$ $$x^a$$ $$\frac{x^{a+1}}{a+1}+C\hbox{ if }a\ne-1$$ $$\frac{1}{x}$$ $$\ln|x|+C$$ $$g(x)^ag'(x)$$ $$\frac{g(x)^{a+1}}{a+1}+C\hbox{ if }a\ne -1$$

 $$f(x)$$ $$F(x)=\int f(x)\ \mathrm{d}{x}$$ $$\sin x$$ $$-\cos x+C$$ $$g'(x)\sin g(x)$$ $$-\cos g(x)+C$$ $$\cos x$$ $$\sin x+C$$ $$\tan x$$ $$\ln|\sec x|+C$$ $$\csc x$$ $$\ln |\csc x-\cot x|+C$$ $$\sec x$$ $$\ln |\sec x+\tan x|+C$$ $$\cot x$$ $$\ln|\sin x|+C$$ $$\sec^2 x$$ $$\tan x+C$$ $$\csc^2 x$$ $$-\cot x+C$$ $$\sec x\tan x$$ $$\sec x+C$$ $$\csc x\cot x$$ $$-\csc x+C$$

 $$f(x)$$ $$F(x)=\int f(x)\ \mathrm{d}{x}$$ $$e^x$$ $$e^x+C$$ $$e^{g(x)}g'(x)$$ $$e^{g(x)}+C$$ $$e^{ax}$$ $$\frac{1}{a}\ e^{ax}+C$$ $$a^x$$ $$\frac{1}{\ln a}\ a^x+C$$ $$\ln x$$ $$x\ln x -x+C$$ $$\frac{1}{\sqrt{1-x^2}}$$ $$\arcsin x+C$$ $$\frac{g'(x)}{\sqrt{1-g(x)^2}}$$ $$\arcsin g(x)+C$$ $$\frac{1}{\sqrt{a^2-x^2}}$$ $$\arcsin \frac{x}{a}+C$$ $$\frac{1}{1+x^2}$$ $$\arctan x+C$$ $$\frac{g'(x)}{1+g(x)^2}$$ $$\arctan g(x)+C$$ $$\frac{1}{a^2+x^2}$$ $$\frac{1}{a}\arctan \frac{x}{a}+C$$ $$\frac{1}{x\sqrt{x^2-1}}$$ $$\textrm{arcsec} x+C$$\ quad ($$x \gt 1$$)

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