Całki funkcji niewymiernych

(A01) $\int \sqrt{a^2-x^2}dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\arcsin \frac{x}{|a|}\text{(}|x|\le |a|\text{)}$

(A02) $\int x\sqrt{a^2-x^2}dx=-\frac{1}{3}\sqrt{(a^2-x^2{)}^3}\text{(}|x|\le |a|\text{)}$

(A03) $\int x^2\sqrt{a^2-x^2}dx=\frac{1}{8}{a}^4\arcsin \frac{x}{|a|}+\frac{1}{8}x(2x^2-a^2)\sqrt{a^2-x^2}\text{(}|x|<|a|\text{)}$

(A04) $\int \frac{\sqrt{a^2-x^2}dx}{x}=\sqrt{a^2-x^2}-a\ln \left|\frac{a+\sqrt{a^2-x^2}}{x}\right|=\sqrt{a^2-x^2}+a\ln \left|\frac{a-\sqrt{a^2-x^2}}{x}\right|\text{(}0<|x|\le |a|\text{)}$

(A05) $\int \frac{\sqrt{a^2-x^2}dx}{x^2}=-\arcsin \frac{x}{|a|}-\frac{\sqrt{a^2-x^2}}{x}\text{(}0<|x|\le |a|\text{)}$

(A06) $\int \frac{dx}{\sqrt{a^2-x^2}}=\arcsin \frac{x}{|a|}\text{(}|x|<|a|\text{)}$

(A07) $\int \frac{xdx}{\sqrt{a^2-x^2}}=-\sqrt{a^2-x^2}\text{(}|x|<|a|\text{)}$

(A08) $\int \frac{x^{2}dx}{\sqrt{a^2-x^2}}=-\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\arcsin \frac{x}{|a|}\text{(}|x|<|a|\text{)}$

(A09) $\int \frac{dx}{x\sqrt{a^2-x^2}}=-\frac{1}{a}\ln \left|\frac{a+\sqrt{a^2-x^2}}{x}\right|=\frac{1}{a}\ln \left|\frac{a-\sqrt{a^2-x^2}}{x}\right|\text{(}0<|x|<|a|\text{)}$

(A10) $\int \frac{dx}{x^2\sqrt{a^2-x^2}}=-\frac{\sqrt{a^2-x^2}}{a^{2}x}\text{(}0<|x|<|a|\text{)}$

(B01) $\int \sqrt{x^2+a^2}dx=\frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}\ln (x+\sqrt{x^2+a^2})=\frac{x}{2}\sqrt{x^2+a^2}-\frac{a^2}{2}\ln (\sqrt{x^2+a^2}-x)$

(B01a)$\int \sqrt{x^2+a^2}dx=\frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\frac{x}{|a|}$

(B02) $\int x\sqrt{x^2+a^2}dx=\frac{1}{3}\sqrt{(x^2+a^2{)}^3}$

(B03) $\int x^2\sqrt{x^2+a^2}dx=\frac{1}{8}x(2x^2+a^2)\sqrt{x^2+a^2}-\frac{a^4}{8}\ln (x+\sqrt{x^2+a^2})$

(B04) $\int \frac{\sqrt{x^2+a^2}dx}{x}=\sqrt{x^2+a^2}-a\ln \left|\frac{a+\sqrt{x^2+a^2}}{x}\right|=\sqrt{x^2+a^2}+a\ln \left|\frac{\sqrt{x^2+a^2}-a}{x}\right|$

(B05) $\int \frac{\sqrt{x^2+a^2}dx}{x^2}=\ln (x+\sqrt{x^2+a^2})-\frac{\sqrt{x^2+a^2}}{x}=-\ln (\sqrt{x^2+a^2}-x)-\frac{\sqrt{x^2+a^2}}{x}$

(B06) $\int \frac{dx}{\sqrt{x^2+a^2}}=\ln (x+\sqrt{x^2+a^2})=-\ln (\sqrt{x^2+a^2}-x)=\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\frac{x}{|a|}$

(B07) $\int \frac{xdx}{\sqrt{x^2+a^2}}=\sqrt{x^2+a^2}$

(B08) $\int \frac{x^{2}dx}{\sqrt{x^2+a^2}}=\frac{x}{2}\sqrt{x^2+a^2}-\frac{a^2}{2}\ln (x+\sqrt{x^2+a^2})=\frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}\ln (\sqrt{x^2+a^2}-x)=\frac{x}{2}\sqrt{x^2+a^2}-\frac{a^2}{2}\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\frac{x}{|a|}$

(B09) $\int \frac{dx}{x\sqrt{x^2+a^2}}=-\frac{1}{a}\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\frac{a}{x}=-\frac{1}{a}\ln \left|\frac{a+\sqrt{x^2+a^2}}{x}\right|=\frac{1}{a}\ln \left|\frac{\sqrt{x^2+a^2}-a}{x}\right|$

(B10) $\int \frac{dx}{x^2\sqrt{x^2+a^2}}=-\frac{\sqrt{x^2+a^2}}{a^{2}x}$

(C01) $\int \sqrt{x^2-a^2}dx=\frac{x}{2}\sqrt{x^2-a^2}-\frac{a^2}{2}\ln |x+\sqrt{x^2-a^2}|=\frac{x}{2}\sqrt{x^2-a^2}+\frac{a^2}{2}\ln |x-\sqrt{x^2-a^2}|=\frac{x}{2}\sqrt{x^2-a^2}-\frac{a^2}{2}\mathrm{sgn}x\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left|\frac{x}{a}\right|\text{(dla }|x|\ge |a|\text{)}$

(C02) $\int x\sqrt{x^2-a^2}dx=\frac{1}{3}\sqrt{(x^2-a^2{)}^3}\text{(dla }|x|\ge |a|\text{)}$

(C03) $\int x^2\sqrt{x^2-a^2}dx=\frac{1}{8}x(2x^2-a^2)\sqrt{x^2-a^2}-\frac{1}{8}{a}^4\ln |x+\sqrt{x^2-a^2}|\text{(dla }|x|\ge |a|\text{)}$

(C04) $\int \frac{\sqrt{x^2-a^2}dx}{x}=\sqrt{x^2-a^2}+a\arcsin \frac{a}{|x|}=\sqrt{x^2-a^2}-a\arccos \frac{a}{|x|}=\sqrt{x^2-a^2}-a\mathrm{arctg}\frac{\sqrt{x^2-a^2}}{a}\text{(dla }|x|\ge |a|\text{)}$

(C05) $\int \frac{\sqrt{x^2-a^2}dx}{x^2}=\ln |x+\sqrt{x^2-a^2}|-\frac{\sqrt{x^2-a^2}}{x}=-\ln |x-\sqrt{x^2-a^2}|-\frac{\sqrt{x^2-a^2}}{x}\text{(dla }|x|\ge |a|\text{)}$

(C06) $\int \frac{dx}{\sqrt{x^2-a^2}}=\ln |x+\sqrt{x^2-a^2}|=-\ln |x-\sqrt{x^2-a^2}|=\mathrm{sgn}x\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left|\frac{x}{a}\right|\text{(dla }|x|>|a|\text{)}$

(C07) $\int \frac{xdx}{\sqrt{x^2-a^2}}=\sqrt{x^2-a^2}\text{(dla }|x|>|a|\text{)}$

(C08) $\int \frac{x^{2}dx}{\sqrt{x^2-a^2}}=\frac{x}{2}\sqrt{x^2-a^2}+\frac{a^2}{2}\ln |x+\sqrt{x^2-a^2}|=\frac{x}{2}\sqrt{x^2-a^2}-\frac{a^2}{2}\ln |x-\sqrt{x^2-a^2}|=\frac{x}{2}\sqrt{x^2-a^2}+\frac{a^2}{2}\mathrm{sgn}x\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left|\frac{x}{a}\right|\text{(dla }|x|>|a|\text{)}$

(C09) $\int \frac{dx}{x\sqrt{x^2-a^2}}=-\frac{1}{a}\arcsin \frac{a}{|x|}=\frac{1}{a}\arccos \frac{a}{|x|}=\frac{1}{a}\mathrm{arctg}\frac{\sqrt{x^2-a^2}}{a}\text{(dla }|x|>|a|\text{)}$

(C10) $\int \frac{dx}{x^2\sqrt{x^2-a^2}}=\frac{\sqrt{x^2-a^2}}{a^{2}x}\text{(dla }|x|>|a|\text{)}$

Następujące wzory są rozwinięciem wzorów (A06), (B06), (C06), dlatego została im nadana numeracja typu (D06x):

(D06a) $\int \frac{dx}{\sqrt{a{x}^2+bx+c}}=\frac{1}{\sqrt{a}}\ln |2\sqrt{a(a{x}^2+bx+c)}+2ax+b|\text{(dla }a>0\text{)}$

(D06b) $\int \frac{dx}{\sqrt{a{x}^2+bx+c}}=\frac{1}{\sqrt{a}}\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\frac{2ax+b}{\sqrt{4ac-b^2}}\text{(dla }a>0\text{, }4ac-b^2>0\text{)}$

(D06c) $\int \frac{dx}{\sqrt{a{x}^2+bx+c}}=\frac{1}{\sqrt{a}}\ln |2ax+b|\text{(dla }a>0\text{, }4ac-b^2=0\text{)}$

(D06d) $\int \frac{dx}{\sqrt{a{x}^2+bx+c}}=-\frac{1}{\sqrt{-a}}\arcsin \frac{2ax+b}{\sqrt{b^2-4ac}}\text{(dla }a<0\text{, }4ac-b^2<0\text{)}$

(D06e) $\int \frac{xdx}{\sqrt{a{x}^2+bx+c}}=\frac{\sqrt{a{x}^2+bx+c}}{a}-\frac{b}{2a}\int \frac{dx}{\sqrt{a{x}^2+bx+c}}$

Uwaga. Wyrażenia podane jako różne wyniki dla danej całki niekoniecznie muszą być równe - mogą się różnić o pewną stałą (przy czym w poszczególnych przedziałach stałe te mogą być różne).

(E01) $\int \sqrt[n]{ax+b}dx=\frac{n}{a(n+1)}\sqrt[n]{{(ax+b)}^{n+1}}+C\text{(dla }n\ge 2\text{)}$

(E02) $\int x\sqrt[n]{ax+b}dx=\frac{n}{a^2(2n+1)}\sqrt[n]{{(ax+b)}^{2n+1}}-\frac{nb}{a^2(n+1)}\sqrt[n]{{(ax+b)}^{n+1}}+C\text{(dla }n\ge 2\text{)}$

(E03) $\int x^2\sqrt[n]{ax+b}dx=\frac{n}{a^3(3n+1)}\sqrt[n]{{(ax+b)}^{3n+1}}-\frac{2nb}{a^3(2n+1)}\sqrt[n]{{(ax+b)}^{2n+1}}+\frac{n{b}^2}{a^3(n+1)}\sqrt[n]{{(ax+b)}^{n+1}}+C\text{(dla }n\ge 2\text{)}$

(E04) $\int x^m\sqrt[n]{ax+b}dx=\frac{n}{a^{m+1}}\left(\sum _{k=0}^m{(-1)}^k\frac{(\genfrac{}{}{0ex}{}{m}{k})b^k}{n(m-k+1)+1}\sqrt[n]{{(ax+b)}^{n(m-k+1)+1}}\right)+C\text{(dla }n\ge 2,m\ge 0\text{)}$

(E05) $\int \frac{dx}{\sqrt[n]{ax+b}}=\frac{n}{a(n-1)}\sqrt[n]{{(ax+b)}^{n-1}}+C\text{(dla }n\ge 2\text{)}$

(E06) $\int \frac{xdx}{\sqrt[n]{ax+b}}=\frac{n}{a^2}(\frac{1}{2n-1}\sqrt[n]{{(ax+b)}^{2n-1}}-\frac{b}{n-1}\sqrt[n]{{(ax+b)}^{n-1}})+C\text{(dla }n\ge 2\text{)}$

(E07)$\int \frac{x^{2}dx}{\sqrt[n]{ax+b}}=\frac{n}{a^3}(\frac{1}{3n-1}\sqrt[n]{{(ax+b)}^{3n-1}}-\frac{2b}{2n-1}\sqrt[n]{{(ax+b)}^{2n-1}}+\frac{b^2}{n-1}\sqrt[n]{{(ax+b)}^{n-1}})+C\text{(dla }n\ge 2\text{)}$

(E08) $\int \frac{x^{m}dx}{\sqrt[n]{ax+b}}=\frac{n}{a^{m+1}}\left(\sum _{k=0}^m{(-1)}^k\frac{(\genfrac{}{}{0ex}{}{m}{k})b^k}{n(m-k+1)-1}\sqrt[n]{{(ax+b)}^{n(m-k+1)-1}}\right)+C\text{(dla }n\ge 2,m\ge 0\text{)}$