Całki funkcji wymiernych

$\int 1\text{d}x=x$

$\int x^n\text{d}x=\frac{x^{n+1}}{n+1}(n\ne -1)$

$\int (ax+b{)}^n\text{d}x=\frac{(ax+b{)}^{n+1}}{a(n+1)}\text{(dla }n\ne -1\text{)}$

$\int \frac{1}{x}\text{d}x=\ln |x|$

$\int \frac{1}{x^n}\text{d}x=\frac{-1}{(n-1)x^{n-1}}(n\ne 1)$

$\int \frac{\text{d}x}{ax+b}=\frac{1}{a}\ln |ax+b|$

$\int x(ax+b{)}^n\text{d}x=\frac{a(n+1)x-b}{a^2(n+1)(n+2)}(ax+b{)}^{n+1}\text{(dla }n\in ̸\{-1,-2\}\text{)}$

$\int \frac{x\text{d}x}{ax+b}=\frac{x}{a}-\frac{b}{a^2}\ln |ax+b|$

$\int \frac{x\text{d}x}{(ax+b{)}^2}=\frac{b}{a^2(ax+b)}+\frac{1}{a^2}\ln |ax+b|$

$\int \frac{x\text{d}x}{(ax+b{)}^n}=\frac{a(1-n)x-b}{a^2(n-1)(n-2)(ax+b{)}^{n-1}}\text{(dla }n\in ̸\{1,2\}\text{)}$

$\int \frac{x^2\text{d}x}{ax+b}=\frac{1}{a^3}(\frac{(ax+b{)}^2}{2}-2b(ax+b)+b^2\ln |ax+b|)$

$\int \frac{x^2\text{d}x}{(ax+b{)}^2}=\frac{1}{a^3}(ax+b-2b\ln |ax+b|-\frac{b^2}{ax+b})$

$\int \frac{x^2\text{d}x}{(ax+b{)}^3}=\frac{1}{a^3}(\ln |ax+b|+\frac{2b}{ax+b}-\frac{b^2}{2(ax+b{)}^2})$

$\int \frac{x^2\text{d}x}{(ax+b{)}^n}=\frac{1}{a^3}(-\frac{1}{(n-3)(ax+b{)}^{n-3}}+\frac{2b}{(n-2)(a+b{)}^{n-2}}-\frac{b^2}{(n-1)(ax+b{)}^{n-1}})\text{(dla }n\in ̸\{1,2,3\}\text{)}$

$\int \frac{\text{d}x}{x(ax+b)}=-\frac{1}{b}\ln \left|\frac{ax+b}{x}\right|$

$\int \frac{\text{d}x}{x^2(ax+b)}=-\frac{1}{bx}+\frac{a}{b^2}\ln \left|\frac{ax+b}{x}\right|$

$\int \frac{\text{d}x}{x^2(ax+b{)}^2}=-a(\frac{1}{b^2(ax+b)}+\frac{1}{a{b}^{2}x}-\frac{2}{b^3}\ln \left|\frac{ax+b}{x}\right|)$

$\int \frac{\text{d}x}{x^2+a^2}=\frac{1}{a}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{g}\frac{x}{a}$

$\int \frac{\text{d}x}{x^2-a^2}=-\frac{1}{a}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{g}\mathrm{h}\frac{x}{a}=\frac{1}{2a}\ln \frac{x-a}{x+a}\text{(dla }|x|>|a|\text{)}$

$\int \frac{\text{d}x}{a{x}^2+bx+c}=\frac{2}{\sqrt{4ac-b^2}}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{g}\frac{2ax+b}{\sqrt{4ac-b^2}}\text{(dla }4ac-b^2>0\text{)}$

$\int \frac{\text{d}x}{a{x}^2+bx+c}=-\frac{2}{\sqrt{b^2-4ac}}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{g}\mathrm{h}\frac{2ax+b}{\sqrt{b^2-4ac}}=\frac{1}{\sqrt{b^2-4ac}}\ln \left|\frac{2ax+b-\sqrt{b^2-4ac}}{2ax+b+\sqrt{b^2-4ac}}\right|\text{(dla }4ac-b^2<0\text{)}$

$\int \frac{x\text{d}x}{a{x}^2+bx+c}=\frac{1}{2a}\ln |a{x}^2+bx+c|-\frac{b}{2a}\int \frac{\text{d}x}{a{x}^2+bx+c}$

$\int \frac{mx+n}{a{x}^2+bx+c}\text{d}x=\frac{m}{2a}\ln |a{x}^2+bx+c|+\frac{2an-bm}{a\sqrt{4ac-b^2}}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{g}\frac{2ax+b}{\sqrt{4ac-b^2}}\text{(dla }4ac-b^2>0\text{)}$

$\int \frac{mx+n}{a{x}^2+bx+c}\text{d}x=\frac{m}{2a}\ln |a{x}^2+bx+c|+\frac{2an-bm}{a\sqrt{b^2-4ac}}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{g}\mathrm{h}\frac{2ax+b}{\sqrt{b^2-4ac}}\text{(dla }4ac-b^2<0\text{)}$

$\int \frac{\text{d}x}{(a{x}^2+bx+c{)}^n}=\frac{2ax+b}{(n-1)(4ac-b^2)(a{x}^2+bx+c{)}^{n-1}}+\frac{(2n-3)2a}{(n-1)(4ac-b^2)}\int \frac{\text{d}x}{(a{x}^2+bx+c{)}^{n-1}}$

$\int \frac{x\text{d}x}{(a{x}^2+bx+c{)}^n}=\frac{bx+2c}{(n-1)(4ac-b^2)(a{x}^2+bx+c{)}^{n-1}}-\frac{b(2n-3)}{(n-1)(4ac-b^2)}\int \frac{\text{d}x}{(a{x}^2+bx+c{)}^{n-1}}$

$\int \frac{\text{d}x}{x(a{x}^2+bx+c)}=\frac{1}{2c}\ln \left|\frac{x^2}{a{x}^2+bx+c}\right|-\frac{b}{2c}\int \frac{\text{d}x}{a{x}^2+bx+c}$

$\int \frac{\text{d}x}{(a+b{x}^n{)}^p}=\frac{x}{(n(p-1)a)(a+b{x}^n{)}^{p-1}}-\frac{1-np+n}{n(p-1)a}\int \frac{\text{d}x}{(a+b{x}^n{)}^{p-1}}$

Szablon:PD-ineligible-tekst