Списак интеграла ирационалних функција

Из пројекта Википедија

Списак интеграла ирационалних функција:

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\int\frac{\sqrt{a^2-x^2}\;dx}{x} = \sqrt{a^2-x^2}-a\ln\left|\frac{a+\sqrt{a^2+x^2}}{x}\right| \qquad\mbox{(}|x|\leq|a|\mbox{)}
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x|\leq|a|\mbox{)}" />
\int\sqrt{x^2+a^2}\;dx = \frac{1}{2}\left(x\sqrt{x^2+a^2}+a^2\,\mathrm{arcsinh}\frac{x}{a}\right)
\int x\sqrt{x^2+a^2}\;dx=\frac13\sqrt{(x^2+a^2)^3}
\int\frac{\sqrt{x^2+a^2}\;dx}{x} = \sqrt{x^2+a^2}-a\ln\left|\frac{a+\sqrt{x^2+a^2}}{x}\right|
x+\sqrt{x^2+a^2}\right|" />
\int\frac{x\,dx}{\sqrt{x^2+a^2}} = \sqrt{x^2+a^2}
\int\frac{x^2\;dx}{\sqrt{x^2+a^2}} = \frac{x}{2}\sqrt{x^2+a^2}-\frac{a^2}{2}\,\mathrm{arcsinh}\frac{x}{a} = \frac{x}{2}\sqrt{x^2+a^2}-\frac{a^2}{2}\ln\left|x+\sqrt{x^2+a^2}\right|
\frac{a+\sqrt{x^2+a^2}}{x}\right|" />
\int\sqrt{x^2-a^2}\;dx = \frac{1}{2}\left(x\sqrt{x^2-a^2}-\sgn x\,\mathrm{arccosh}\left|\frac{x}{a}\right|\right) \qquad\mbox{(for }|x|\ge|a|\mbox{)}
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x|+\sqrt{x^2-a^2}\right) \qquad\mbox{(for }|x|>|a|\mbox{)}" />
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\frac{x}{a}\right| = \frac{1}{2}\left(x\sqrt{x^2-a^2}+a^2\ln\left(|x|+\sqrt{x^2-a^2}\right)\right) \qquad\mbox{(for }|x|>|a|\mbox{)}" />
2\sqrt{a(ax^2+bx+c)}+2ax+b\right| \qquad\mbox{(for }a>0\mbox{)}" />
\int\frac{dx}{\sqrt{ax^2+bx+c}} = \frac{1}{\sqrt{a}}\,\mathrm{arcsinh}\frac{2ax+b}{\sqrt{4ac-b^2}} \qquad\mbox{(for }a>0\mbox{, }4ac-b^2>0\mbox{)}
2ax+b| \qquad\mbox{(for }a>0\mbox{, }4ac-b^2=0\mbox{)}" />
\int\frac{dx}{\sqrt{ax^2+bx+c}} = -\frac{1}{\sqrt{-a}}\arcsin\frac{2ax+b}{\sqrt{b^2-4ac}} \qquad\mbox{(for }a<0\mbox{, }4ac-b^2<0\mbox{)}
\int\frac{x\;dx}{\sqrt{ax^2+bx+c}} = \frac{\sqrt{ax^2+bx+c}}{a}-\frac{b}{2a}\int\frac{dx}{\sqrt{ax^2+bx+c}}