Cesko.wiki Seznam integrálů racionálních funkcí
Technology
12 hours ago
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Albert FloresToto je seznam integrálů (primitivních funkcí) racionálních funkcí.
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| \int (ax + b)^n \mathrm{d}x | = \frac{(ax + b)^{n+1}}{a(n + 1)} \qquad\mbox{(pro } n\neq -1\mbox{)}\,\! |
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| \int\frac{\mathrm{d}x}{ax + b} | = \frac{1}{a}\ln\left|ax + b\right |
| \int x(ax + b)^n \mathrm{d}x | = \frac{a(n + 1)x - b}{a^2(n + 1)(n + 2)} (ax + b)^{n+1} \qquad\mbox{(pro }n \not\in \{-1, -2\}\mbox{)} |
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| \int\frac{x \,\mathrm{d}x}{ax + b} | = \frac{x}{a} - \frac{b}{a^2}\ln\left|ax + b\right |
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| \int\frac{x \,\mathrm{d}x}{(ax + b)^2} | = \frac{b}{a^2(ax + b)} + \frac{1}{a^2}\ln\left|ax + b\right |
| \int\frac{x \,\mathrm{d}x}{(ax + b)^n} | = \frac{a(1 - n)x - b}{a^2(n - 1)(n - 2)(ax + b)^{n-1}} \qquad\mbox{(pro } n\not\in \{1, 2\}\mbox{)} |
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| \int\frac{x^2 \mathrm{d}x}{ax + b} | = \frac{1}{a^3}\left(\frac{(ax + b)^2}{2} - 2b(ax + b) + b^2\ln\left|ax + b\right|\right) |
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| \int\frac{x^2 \mathrm{d}x}{(ax + b)^2} | = \frac{1}{a^3}\left(ax + b - 2b\ln\left|ax + b\right| - \frac{b^2}{ax + b}\right) |
| \int\frac{x^2 \mathrm{d}x}{(ax + b)^3} | = \frac{1}{a^3}\left(\ln\left|ax + b\right| + \frac{2b}{ax + b} - \frac{b^2}{2(ax + b)^2}\right) |
| \int\frac{x^2 \mathrm{d}x}{(ax + b)^n} | = \frac{1}{a^3}\left(-\frac{(ax + b)^{3-n}}{(n-3)} + \frac{2b (a + b)^{2-n}}{(n-2)} - \frac{b^2 (ax + b)^{1-n}}{(n - 1)}\right) \qquad\mbox{(pro } n\not\in \{1, 2, 3\}\mbox{)} |
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| \int\frac{\mathrm{d}x}{x(ax + b)} | = -\frac{1}{b}\ln\left|\frac{ax+b}{x}\right |
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| \int\frac{\mathrm{d}x}{x^2(ax+b)} | = -\frac{1}{bx} + \frac{a}{b^2}\ln\left|\frac{ax+b}{x}\right |
| \int\frac{\mathrm{d}x}{x^2(ax+b)^2} | = -a\left(\frac{1}{b^2(ax+b)} + \frac{1}{ab^2x} - \frac{2}{b^3}\ln\left|\frac{ax+b}{x}\right|\right) |
| \int\frac{\mathrm{d}x}{x^2+a^2} | = \frac{1}{a}\arctan\frac{x}{a}\,\. |
| \int\frac{\mathrm{d}x}{x^2-a^2} = | -\frac{1}{a}\,\mathrm{arctanh}\frac{x}{a} = \frac{1}{2a}\ln\frac{a-x}{a+x} \qquad\mbox{(pro }|x |
| -\frac{1}{a}\,\mathrm{arccoth}\frac{x}{a} = \frac{1}{2a}\ln\frac{x-a}{x+a} \qquad\mbox{(pro }|x| > |a|\mbox{)}\,\. +more |
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| \int\frac{\mathrm{d}x}{ax^2+bx+c} = | \frac{2}{\sqrt{4ac-b^2}}\arctan\frac{2ax+b}{\sqrt{4ac-b^2}} \qquad\mbox{(pro }4ac-b^2>0\mbox{)} |
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| \frac{2}{\sqrt{b^2-4ac}}\,\mathrm{artanh}\frac{2ax+b}{\sqrt{b^2-4ac}} = \frac{1}{\sqrt{b^2-4ac}}\ln\left|\frac{2ax+b-\sqrt{b^2-4ac}}{2ax+b+\sqrt{b^2-4ac}}\right| \qquad\mbox{(pro }4ac-b^2 | |
| -\frac{2}{2ax+b}\qquad\mbox{(pro }4ac-b^2=0\mbox{)} | |
| \int\frac{x \,\mathrm{d}x}{ax^2+bx+c} | = \frac{1}{2a}\ln\left|ax^2+bx+c\right |
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| \frac{m}{2a}\ln\left|ax^2+bx+c\right |
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: \int\frac{\mathrm{d}x}{(ax^2+bx+c)^n} = \frac{2ax+b}{(n-1)(4ac-b^2)(ax^2+bx+c)^{n-1}}+\frac{(2n-3)2a}{(n-1)(4ac-b^2)}\int\frac{\mathrm{d}x}{(ax^2+bx+c)^{n-1}}\,\. : \int\frac{x \,\mathrm{d}x}{(ax^2+bx+c)^n} = \frac{bx+2c}{(n-1)(4ac-b^2)(ax^2+bx+c)^{n-1}}-\frac{b(2n-3)}{(n-1)(4ac-b^2)}\int\frac{\mathrm{d}x}{(ax^2+bx+c)^{n-1}}\,\. +more : \int\frac{\mathrm{d}x}{x(ax^2+bx+c)} = \frac{1}{2c}\ln\left|\frac{x^2}{ax^2+bx+c}\right|-\frac{b}{2c}\int\frac{\mathrm{d}x}{ax^2+bx+c}.
Jakoukoliv racionální funkci lze integrovat výše uvedenými rovnicemi a metodou rozkladu na parciální zlomky, rozkladem racionální funkce na sumu výrazů ve tvaru: : \frac{ex + f}{\left(ax^2+bx+c\right)^n}.