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Section A.4 Table of Integrals
Throughout this table, \(a\) and \(b\) are given constants, independent of \(x\) and \(C\) is an arbitrary constant.
| \(f(x)\) |
\(F(x)=\int f(x)\ \dee{x}\) |
| \(af(x)+bg(x)\) |
\(a\int f(x)\ \dee{x}+b\int g(x)\ \dee{x}\ +\ C\) |
| \(f(x)+g(x)\) |
\(\int f(x)\ \dee{x}+\int g(x)\ \dee{x}\ +\ C\) |
| \(f(x)-g(x)\) |
\(\int f(x)\ \dee{x}-\int g(x)\ \dee{x}\ +\ C\) |
| \(af(x)\) |
\(a\int f(x)\ \dee{x}\ +\ C\) |
| \(u(x)v'(x)\) |
\(u(x)v(x)-\int u'(x)v(x)\ \dee{x}\ +\ C\) |
| \(f\big(y(x)\big)y'(x)\) |
\(F\big(y(x)\big)\hbox{ where }F(y)=\int f(y)\ \dee{y}\) |
| \(a\) |
\(ax+C\) |
| \(x^a\) |
\(\frac{x^{a+1}}{a+1}+C\hbox{ if }a\ne-1\) |
| \(\frac{1}{x}\) |
\(\ln|x|+C\) |
| \(g(x)^ag'(x)\) |
\(\frac{g(x)^{a+1}}{a+1}+C\hbox{ if }a\ne -1\) |
| \(f(x)\) |
\(F(x)=\int f(x)\ \dee{x}\) |
| \(\sin x\) |
\(-\cos x+C\) |
| \(g'(x)\sin g(x)\) |
\(-\cos g(x)+C\) |
| \(\cos x\) |
\(\sin x+C\) |
| \(\tan x\) |
\(\ln|\sec x|+C\) |
| \(\csc x\) |
\(\ln |\csc x-\cot x|+C\) |
| \(\sec x\) |
\(\ln |\sec x+\tan x|+C\) |
| \(\cot x\) |
\(\ln|\sin x|+C\) |
| \(\sec^2 x\) |
\(\tan x+C\) |
| \(\csc^2 x\) |
\(-\cot x+C\) |
| \(\sec x\tan x\) |
\(\sec x+C\) |
| \(\csc x\cot x\) |
\(-\csc x+C\) |
| \(f(x)\) |
\(F(x)=\int f(x)\ \dee{x}\) |
| \(e^x\) |
\(e^x+C\) |
| \(e^{g(x)}g'(x)\) |
\(e^{g(x)}+C\) |
| \(e^{ax}\) |
\(\frac{1}{a}\ e^{ax}+C\) |
| \(a^x\) |
\(\frac{1}{\ln a}\ a^x+C\) |
| \(\ln x\) |
\(x\ln x -x+C\) |
| \(\frac{1}{\sqrt{1-x^2}}\) |
\(\arcsin x+C\) |
| \(\frac{g'(x)}{\sqrt{1-g(x)^2}}\) |
\(\arcsin g(x)+C\) |
| \(\frac{1}{\sqrt{a^2-x^2}}\) |
\(\arcsin \frac{x}{a}+C\) |
| \(\frac{1}{1+x^2}\) |
\(\arctan x+C\) |
| \(\frac{g'(x)}{1+g(x)^2}\) |
\(\arctan g(x)+C\) |
| \(\frac{1}{a^2+x^2}\) |
\(\frac{1}{a}\arctan \frac{x}{a}+C\) |
| \(\frac{1}{x\sqrt{x^2-1}}\) |
\(\arcsec x+C\hbox{ for }x \gt 1\) |
