A.4: Таблиця інтегралів
- Page ID
- 60939
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По всій цій таблиці\(a\) і\(b\) задані константи, незалежні від\(x\) і\(C\) є довільною константою.
| \(f(x)\) | \(F(x)=\int f(x)\ \text{d}x\) |
| \(af(x)+bg(x)\) | \(a\int f(x)\ \text{d}x+b\int g(x)\ \text{d}x\ +\ C\) |
| \(f(x)+g(x)\) | \(\int f(x)\ \text{d}x+\int g(x)\ \text{d}x\ +\ C\) |
| \(f(x)-g(x)\) | \(\int f(x)\ \text{d}x-\int g(x)\ \text{d}x\ +\ C\) |
| \(af(x)\) | \(a\int f(x)\ \text{d}x\ +\ C\) |
| \(u(x)v'(x)\) | \(u(x)v(x)-\int u'(x)v(x)\ \text{d}x\ +\ C\) |
| \(f\big(y(x)\big)y'(x)\) | \(F\big(y(x)\big)\hbox{ where }F(y)=\int f(y)\ \text{d}y\) |
| \(a\) | \(ax+C\) |
| \(x^a\) | \(\frac{x^{a+1}}{a+1}+C\hbox{ if }a\ne-1\) |
| \(\frac{1}{x}\) | \(\ln|x|+C\) |
| \(g(x)^ag'(x)\) | \(\frac{g(x)^{a+1}}{a+1}+C\hbox{ if }a\ne -1\) |
| \(f(x)\) | \(F(x)=\int f(x)\ \text{d}x\) |
| \(\sin x\) | \(-\cos x+C\) |
| \(g'(x)\sin g(x)\) | \(-\cos g(x)+C\) |
| \(\cos x\) | \(\sin x+C\) |
| \(\tan x\) | \(\ln|\sec x|+C\) |
| \(\csc x\) | \(\ln |\csc x-\cot x|+C\) |
| \(\sec x\) | \(\ln |\sec x+\tan x|+C\) |
| \(\cot x\) | \(\ln|\sin x|+C\) |
| \(\sec^2 x\) | \(\tan x+C\) |
| \(\csc^2 x\) | \(-\cot x+C\) |
| \(\sec x\tan x\) | \(\sec x+C\) |
| \(\csc x\cot x\) | \(-\csc x+C\) |
| \(f(x)\) | \(F(x)=\int f(x)\ \text{d}x\) |
| \(e^x\) | \(e^x+C\) |
| \(e^{g(x)}g'(x)\) | \(e^{g(x)}+C\) |
| \(e^{ax}\) | \(\frac{1}{a}\ e^{ax}+C\) |
| \(a^x\) | \(\frac{1}{\ln a}\ a^x+C\) |
| \(\ln x\) | \(x\ln x -x+C\) |
| \(\frac{1}{\sqrt{1-x^2}}\) | \(\arcsin x+C\) |
| \(\frac{g'(x)}{\sqrt{1-g(x)^2}}\) | \(\arcsin g(x)+C\) |
| \(\frac{1}{\sqrt{a^2-x^2}}\) | \(\arcsin \frac{x}{a}+C\) |
| \(\frac{1}{1+x^2}\) | \(\arctan x+C\) |
| \(\frac{g'(x)}{1+g(x)^2}\) | \(\arctan g(x)+C\) |
| \(\frac{1}{a^2+x^2}\) | \(\frac{1}{a}\arctan \frac{x}{a}+C\) |
| \(\frac{1}{x\sqrt{x^2-1}}\) | \(\textrm{arcsec} x+C\)\ квадроцикл (\(x \gt 1\)) |