Let \(k, n\), and \(a\) be any real number (constants). Assume that \(a > 0\) and that \(a\) for formulas 6, 8-11 and, and 13. The integration constant is denoted as \(C\).
\(\int kf(x)dx=k\int f(x)dx\)
\(\int[f(x)+g(x)]dx=\int f(x)dx+\int g(x)dx\)
\(\int kdx=kx+C\)
\(\int x^ndx=\dfrac{x^{n+1}}{n+1}+C,\quad n\neq-1\)
\(\int\frac{1}{x}dx=\int x^{-1}dx=\ln\mid x\mid+C\)
\(\int e^{kx}dx=\frac{e^{kx}}{k}+C\)
\(\int a^x=\frac{a^x}{\ln a}+C\)
\(\int\cos(kx)dx=\frac{\sin(kx)}{k}+C\)
\(\int\sin(kx)dx=-\frac{\cos(kx)}{k}+C\)
\(\int\sec^2(kx)dx=\frac{\tan(kx)}{k}+C\)
\(\int\sec(kx)\tan(kx)dx=\frac{\sec(kx)}{k}+C\)
\(\int\sec xdx=\ln\lvert\sec x+\tan x\rvert+C\)
\(\int\tan(kx)dx=\frac{\ln\left|\sec(kx)\right|}{k}+C\)
\(\int\frac{1}{\sqrt{a^2-x^2}}dx=\frac{1}{a}\arcsin\left(\frac{x}{a}\right)+C\)
\(\int\frac{-1}{\sqrt{a^2-x^2}}dx=-\frac{1}{a}\arccos\left(\frac{x}{a}\right)+C\)
- \(\int\frac{1}{x^2+a^2}dx=\frac{1}{a}\arctan\left(\frac{x}{a}\right)+C\)