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2021-08-23T07:22:31Z

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Slijedi popis [[integral]]a ([[antiderivacija]] funkcija) [[racionalna funkcija|racionalnih funkcija]]. Za potpun popis integrala funkcija, pogledati [[tablica integrala]] i [[Popisi integrala|popis integrala]].
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|<math>\int (ax + b)^n dx</math> || <math> = \frac{(ax + b)^{n+1}}{a(n + 1)} + C \qquad\mbox{(za } n\neq -1\mbox{)}\,\!</math>
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|<math>\int\frac{1}{ax + b} dx</math> || <math> = \frac{1}{a}\ln\left|ax + b\right| + C</math>
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|<math>\int x(ax + b)^n dx</math> || <math> = \frac{a(n + 1)x - b}{a^2(n + 1)(n + 2)} (ax + b)^{n+1} + C \qquad\mbox{(za }n \not\in \{-1, -2\}\mbox{)}</math>
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:{|
|<math>\int\frac{x}{ax + b} dx</math> || <math> = \frac{x}{a} - \frac{b}{a^2}\ln\left|ax + b\right| + C</math>
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|<math>\int\frac{x}{(ax + b)^2} dx</math>||<math> = \frac{b}{a^2(ax + b)} + \frac{1}{a^2}\ln\left|ax + b\right| + C</math>
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|<math>\int\frac{x}{(ax + b)^n} dx</math>||<math> = \frac{a(1 - n)x - b}{a^2(n - 1)(n - 2)(ax + b)^{n-1}} + C \qquad\mbox{(za } n\not\in \{1, 2\}\mbox{)}</math>
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:{|
|<math>\int\frac{x^2}{ax + b} dx</math>||<math> = \frac{1}{a^3}\left(\frac{(ax + b)^2}{2} - 2b(ax + b) + b^2\ln\left|ax + b\right|\right) + C</math>
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|<math>\int\frac{x^2}{(ax + b)^2} dx</math>||<math> = \frac{1}{a^3}\left(ax + b - 2b\ln\left|ax + b\right| - \frac{b^2}{ax + b}\right) + C</math>
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|<math>\int\frac{x^2}{(ax + b)^3} dx</math>||<math> = \frac{1}{a^3}\left(\ln\left|ax + b\right| + \frac{2b}{ax + b} - \frac{b^2}{2(ax + b)^2}\right) + C</math>
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|<math>\int\frac{x^2}{(ax + b)^n} dx</math>||<math> = \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) + C \qquad\mbox{(za } n\not\in \{1, 2, 3\}\mbox{)}</math>
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:{|
|<math>\int\frac{1}{x(ax + b)} dx</math>||<math> = -\frac{1}{b}\ln\left|\frac{ax+b}{x}\right| + C</math>
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|<math>\int\frac{1}{x^2(ax+b)} dx</math>||<math> = -\frac{1}{bx} + \frac{a}{b^2}\ln\left|\frac{ax+b}{x}\right| + C</math>
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|<math>\int\frac{1}{x^2(ax+b)^2} dx</math>||<math> = -a\left(\frac{1}{b^2(ax+b)} + \frac{1}{ab^2x} - \frac{2}{b^3}\ln\left|\frac{ax+b}{x}\right|\right) + C</math>
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|<math>\int\frac{1}{x^2+a^2} dx</math>||<math> = \frac{1}{a}\arctan\frac{x}{a}\,\! + C</math>
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|<math>\int\frac{1}{x^2-a^2} dx = </math>||
<math>\begin{cases}
-\frac{1}{a}\,\mathrm{arctanh}\frac{x}{a} = \frac{1}{2a}\ln\frac{a-x}{a+x} + C \qquad\mbox{(za }|x| < |a|\mbox{)}\,\! \\
-\frac{1}{a}\,\mathrm{arccoth}\frac{x}{a} = \frac{1}{2a}\ln\frac{x-a}{x+a} + C \qquad\mbox{(za }|x| > |a|\mbox{)}\,\!
\end{cases}</math>
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:{|
|<math>\int\frac{1}{ax^2+bx+c} dx =</math>||
<math>\begin{cases}
\frac{2}{\sqrt{4ac-b^2}}\arctan\frac{2ax+b}{\sqrt{4ac-b^2}} + C \qquad\mbox{(za }4ac-b^2>0\mbox{)} \\
-\frac{2}{\sqrt{b^2-4ac}}\,\mathrm{arctanh}\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| + C \qquad\mbox{(za }4ac-b^2<0\mbox{)} \\
-\frac{2}{2ax+b}+ C \qquad\mbox{(za }4ac-b^2=0\mbox{)}
\end{cases}</math>
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|<math>\int\frac{x}{ax^2+bx+c} dx</math>||<math> = \frac{1}{2a}\ln\left|ax^2+bx+c\right|-\frac{b}{2a}\int\frac{dx}{ax^2+bx+c} + C</math>
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:{|
|<math>\int\frac{mx+n}{ax^2+bx+c} dx = </math> ||
<math>\begin{cases}
\frac{m}{2a}\ln\left|ax^2+bx+c\right|+\frac{2an-bm}{a\sqrt{4ac-b^2}}\arctan\frac{2ax+b}{\sqrt{4ac-b^2}} + C \qquad\mbox{(za }4ac-b^2>0\mbox{)} \\
\frac{m}{2a}\ln\left|ax^2+bx+c\right|+\frac{2an-bm}{a\sqrt{b^2-4ac}}\,\mathrm{artanh}\frac{2ax+b}{\sqrt{b^2-4ac}} + C \qquad\mbox{(za }4ac-b^2<0\mbox{)} \\
\frac{m}{2a}\ln\left|ax^2+bx+c\right|-\frac{2an-bm}{a(2ax+b)} + C \,\,\,\,\,\,\,\,\,\, \qquad\mbox{(za }4ac-b^2=0\mbox{)}
\end{cases}</math>
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: <math>\int\frac{1}{(ax^2+bx+c)^n} dx= \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{1}{(ax^2+bx+c)^{n-1}} dx\,\! + C</math>
: <math>\int\frac{x}{(ax^2+bx+c)^n} dx= \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{1}{(ax^2+bx+c)^{n-1}} dx\,\! + C</math>
: <math>\int\frac{1}{x(ax^2+bx+c)} dx= \frac{1}{2c}\ln\left|\frac{x^2}{ax^2+bx+c}\right|-\frac{b}{2c}\int\frac{1}{ax^2+bx+c} dx + C</math>

Bilo koja racionalna funkcija se može integrirati rabeći gornje jednadžbe i '''[[parcijalni razlomci u integriranju|parcijalne razlomke u integriranju]]''', dekompozicijom racionalne funkcije u zbroj funkcija oblika:
: <math>\frac{ex + f}{\left(ax^2+bx+c\right)^n}</math>.

{{Popisi integrala}}

[[Kategorija:Integrali|Racionalne funkcije]]
[[Kategorija:Matematički popisi|Integrali, racionalne funkcije]]

Popis integrala racionalnih funkcija - Povijest promjena

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