Tavola degli integrali indefiniti di funzioni irrazionali

Questa pagina contiene una tavola di integrali indefiniti di funzioni irrazionali. Per altri integrali vedi Tavole di Integrali.

$\int \sqrt{a^2-x^2}\mathrm{d}x=\frac{1}{2}(x\sqrt{a^2-x^2}+a^2\arcsin \frac{x}{a})\text{(}|x|\le |a|\text{)}$

$\int x\sqrt{a^2-x^2}\mathrm{d}x=-\frac{1}{3}\sqrt{(a^2-x^2{)}^3}\text{(}|x|\le |a|\text{)}$

$\int \frac{\sqrt{a^2-x^2}\mathrm{d}x}{x}=\sqrt{a^2-x^2}-a\log \left|\frac{a+\sqrt{a^2+x^2}}{x}\right|\text{(}|x|\le |a|\text{)}$

$\int \frac{\mathrm{d}x}{\sqrt{a^2-x^2}}=\arcsin \frac{x}{a}\text{(}|x|\le |a|\text{)}$

$\int \frac{x^2\mathrm{d}x}{\sqrt{a^2-x^2}}=-\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\arcsin \frac{x}{a}\text{(}|x|\le |a|\text{)}$

$\int \sqrt{x^2+a^2}\mathrm{d}x=\frac{1}{2}(x\sqrt{x^2+a^2}+a^2\log (x+\sqrt{x^2+a^2}))=\frac{1}{2}(x\sqrt{x^2+a^2}+a^2\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\frac{x}{a})$

$\int x\sqrt{x^2+a^2}\mathrm{d}x=\frac{1}{3}\sqrt{(x^2+a^2{)}^3}$

$\int \frac{\sqrt{x^2+a^2}\mathrm{d}x}{x}=\sqrt{x^2+a^2}-a\log \left|\frac{a+\sqrt{x^2+a^2}}{x}\right|$

$\int \frac{\mathrm{d}x}{\sqrt{x^2+a^2}}=\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\frac{x}{a}=\log |x+\sqrt{x^2+a^2}|$

$\int \frac{x\mathrm{d}x}{\sqrt{x^2+a^2}}=\sqrt{x^2+a^2}$

$\int \frac{x^2\mathrm{d}x}{\sqrt{x^2+a^2}}=\frac{x}{2}\sqrt{x^2+a^2}-\frac{a^2}{2}\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\frac{x}{a}=\frac{x}{2}\sqrt{x^2+a^2}-\frac{a^2}{2}\log |x+\sqrt{x^2+a^2}|$

$\int \frac{\mathrm{d}x}{x\sqrt{x^2+a^2}}=-\frac{1}{a}\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\frac{a}{x}=-\frac{1}{a}\log \left|\frac{a+\sqrt{x^2+a^2}}{x}\right|$

$\int \sqrt{x^2-a^2}\mathrm{d}x=\frac{1}{2}(x\sqrt{x^2-a^2}\mp a^2\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left|\frac{x}{a}\right|)\text{(per }|x|\ge |a|\text{; }-\text{ per }x>0\text{, }+\text{ per }x<0\text{)}$

$\int x\sqrt{x^2-a^2}\mathrm{d}x=\frac{1}{3}\sqrt{(x^2-a^2{)}^3}\text{(per }|x|\ge |a|\text{)}$

$\int \frac{\sqrt{x^2-a^2}\mathrm{d}x}{x}=\sqrt{x^2-a^2}-a\arcsin \frac{a}{x}\text{(per }|x|\ge |a|\text{)}$

$\int \frac{\mathrm{d}x}{\sqrt{x^2-a^2}}=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\frac{x}{a}=\log (|x|+\sqrt{x^2-a^2})\text{(per }|x|>|a|\text{)}$

$\int \frac{x\mathrm{d}x}{\sqrt{x^2-a^2}}=\sqrt{x^2-a^2}\text{(per }|x|>|a|\text{)}$

$\int \frac{x^2\mathrm{d}x}{\sqrt{x^2-a^2}}=\frac{x}{2}\sqrt{x^2-a^2}+\frac{a^2}{2}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left|\frac{x}{a}\right|=\frac{1}{2}(x\sqrt{x^2-a^2}+a^2\ln (|x|+\sqrt{x^2-a^2}))\text{(per }|x|>|a|\text{)}$

$\int \frac{\mathrm{d}x}{\sqrt{a{x}^2+bx+c}}=\frac{1}{\sqrt{a}}\ln |2\sqrt{a(a{x}^2+bx+c)}+2ax+b|\text{(per }a>0\text{)}$

$\int \frac{\mathrm{d}x}{\sqrt{a{x}^2+bx+c}}=\frac{1}{\sqrt{a}}\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\frac{2ax+b}{\sqrt{4ac-b^2}}\text{(per }a>0\text{, }4ac-b^2>0\text{)}$

$\int \frac{\mathrm{d}x}{\sqrt{a{x}^2+bx+c}}=\frac{1}{\sqrt{a}}\log |2ax+b|\text{(per }a>0\text{, }4ac-b^2=0\text{)}$

$\int \frac{\mathrm{d}x}{\sqrt{a{x}^2+bx+c}}=-\frac{1}{\sqrt{-a}}\arcsin \frac{2ax+b}{\sqrt{b^2-4ac}}\text{(per }a<0\text{, }4ac-b^2<0\text{)}$

$\int \frac{x\mathrm{d}x}{\sqrt{a{x}^2+bx+c}}=\frac{\sqrt{a{x}^2+bx+c}}{a}-\frac{b}{2a}\int \frac{\mathrm{d}x}{\sqrt{a{x}^2+bx+c}}$

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