Vytořil jsem tabulku pro výpočet neurčitých integrálů. Tabulka by měla být vhodná jak pro středoškoláky, tak pro vysokoškoláky.
Můžete také stahnout ve formátu pdf zde.
| \(\int f(x){\rm d}x\) | \(F(x)\) | podmínky |
|---|---|---|
| \(\int(x+a)^n{\rm d}x\) | \(\frac{(x + a)^{n + 1}}{n + 1} + C\) | \(n\in\mathbb{Z}\setminus\{-1\}\) \(x \neq -a\) pro \(n < -1\) |
| \(\int x^\alpha{\rm d}x\) | \(\frac{x^{\alpha + 1}}{\alpha + 1} + C\) | \(\alpha \in \mathbb{R} \setminus \{-1\}; x > 0\) |
| \(\int\frac{1}{x + a}{\rm d}x\) | \({\rm ln}|x+a| + C\) | \(a \in \mathbb{R};x \in \mathbb{R} \setminus \{-a\}\) |
| \(\int{\rm ln}(x){\rm d}x\) | \(-x+x\,{\rm ln}(x)+C\) | \(x\in(0,+\infty)\) |
| \(\int e^x{\rm d}x\) | \(e^x + C\) | \(x \in \mathbb{R}\) |
| \(\int\sin(x){\rm d}x\) | \( – \cos(x) + C\) | |
| \(\int\cos(x){\rm d}x\) | \(\sin(x) + C\) | |
| \(\int\frac{1}{1 + x^2}{\rm d}x\) | \(\arctan(x) + C_1\) | |
| \( – {\rm arccot}(x)+C_2\) | ||
| \(\int\frac{1}{\sqrt{1 – x^2}}{\rm d}x\) |
\({\rm arcsin}(x) + C_1\) | \(x \in ( -1, 1)\) |
| \({\rm arccos}(x) + C_2\) | ||
| \(\int\frac{1}{\sqrt{1 + x^2}}{\rm d}x\) |
\({\rm argsinh}(x) + C_1\) | \(x \in \mathbb{R}\) |
| \({\rm ln}\left(x + \sqrt{x^2 + 1}\right) + C_2\) | ||
| \(\int\frac{1}{\sqrt{x^2 – 1}}{\rm d}x\) |
\({\rm argcosh}(x) + C_1\) | \(|x| > 1\) |
| \({\rm ln}\left(x + \sqrt{x^2 – 1}\right) + C_2\) | ||
| \(\int\cosh(x){\rm d}x\) | \(\sinh(x) + C\) | \(x \in \mathbb{R}\) |
| \(\int\sinh(x){\rm d}x\) | \(\cos(x) + C\) | |
| \(\int\frac{1}{\cos^2(x)}{\rm d}x\) | \(\tan(x) + C\) | \(x \in \left( – \frac{\pi}{2} + k\,\pi, \frac{\pi}{2} + k\,\pi \right); k \in \mathbb{Z} \) |
| \(\int\frac{1}{\sin^2(x)}{\rm d}x\) | \( – \cot(x) + C\) | \(x \in ( k\,\pi, (k+1)\,\pi ); k \in \mathbb{Z} \) |