Tích phân hàm số toán cotangent

$\int \cot axdx=\frac{1}{a}\ln |\sin ax|+C$

$\int {\cot }^{n}axdx=-\frac{1}{a(n-1)}{\cot }^{n-1}ax-\int {\cot }^{n-2}axdx\text{(for }n\ne 1\text{)}$

$\int \frac{dx}{1+\cot ax}=\int \frac{\tan axdx}{\tan ax+1}$

$\int \frac{dx}{1-\cot ax}=\int \frac{\tan axdx}{\tan ax-1}$

$\int \frac{dx}{\cos ax\pm \sin ax}=\frac{1}{a\sqrt{2}}\ln |\tan (\frac{ax}{2}\pm \frac{\pi }{8})|+C$

$\int \frac{dx}{(\cos ax\pm \sin ax{)}^2}=\frac{1}{2a}\tan (ax\mp \frac{\pi }{4})+C$

$\int \frac{dx}{(\cos x+\sin x{)}^n}=\frac{1}{n-1}(\frac{\sin x-\cos x}{(\cos x+\sin x{)}^{n-1}}-2(n-2)\int \frac{dx}{(\cos x+\sin x{)}^{n-2}})$

$\int \frac{\cos axdx}{\cos ax+\sin ax}=\frac{x}{2}+\frac{1}{2a}\ln |\sin ax+\cos ax|+C$

$\int \frac{\cos axdx}{\cos ax-\sin ax}=\frac{x}{2}-\frac{1}{2a}\ln |\sin ax-\cos ax|+C$

$\int \frac{\sin axdx}{\cos ax+\sin ax}=\frac{x}{2}-\frac{1}{2a}\ln |\sin ax+\cos ax|+C$

$\int \frac{\sin axdx}{\cos ax-\sin ax}=-\frac{x}{2}-\frac{1}{2a}\ln |\sin ax-\cos ax|+C$

$\int \frac{\cos axdx}{\sin ax(1+\cos ax)}=-\frac{1}{4a}{\tan }^2\frac{ax}{2}+\frac{1}{2a}\ln |\tan \frac{ax}{2}|+C$

$\int \frac{\cos axdx}{\sin ax(1+-\cos ax)}=-\frac{1}{4a}{\cot }^2\frac{ax}{2}-\frac{1}{2a}\ln |\tan \frac{ax}{2}|+C$

$\int \frac{\sin axdx}{\cos ax(1+\sin ax)}=\frac{1}{4a}{\cot }^2(\frac{ax}{2}+\frac{\pi }{4})+\frac{1}{2a}\ln |\tan (\frac{ax}{2}+\frac{\pi }{4})|+C$

$\int \frac{\sin axdx}{\cos ax(1-\sin ax)}=\frac{1}{4a}{\tan }^2(\frac{ax}{2}+\frac{\pi }{4})-\frac{1}{2a}\ln |\tan (\frac{ax}{2}+\frac{\pi }{4})|+C$

$\int \sin ax\cos axdx=\frac{1}{2a}{\sin }^{2}ax+C$

$\int \sin a_{1}x\cos a_{2}xdx=-\frac{\cos (a_1+a_2)x}{2(a_1+a_2)}-\frac{\cos (a_1-a_2)x}{2(a_1-a_2)}+C\text{(for }|a_1|\ne |a_2|\text{)}$

$\int {\sin }^{n}ax\cos axdx=\frac{1}{a(n+1)}{\sin }^{n+1}ax+C\text{(for }n\ne -1\text{)}$

$\int \sin ax{\cos }^{n}axdx=-\frac{1}{a(n+1)}{\cos }^{n+1}ax+C\text{(for }n\ne -1\text{)}$

$\int {\sin }^{n}ax{\cos }^{m}axdx=-\frac{{\sin }^{n-1}ax{\cos }^{m+1}ax}{a(n+m)}+\frac{n-1}{n+m}\int {\sin }^{n-2}ax{\cos }^{m}axdx\text{(for }m,n>0\text{)}$

also: $\int {\sin }^{n}ax{\cos }^{m}axdx=\frac{{\sin }^{n+1}ax{\cos }^{m-1}ax}{a(n+m)}+\frac{m-1}{n+m}\int {\sin }^{n}ax{\cos }^{m-2}axdx\text{(for }m,n>0\text{)}$

$\int \frac{dx}{\sin ax\cos ax}=\frac{1}{a}\ln |\tan ax|+C$

$\int \frac{dx}{\sin ax{\cos }^{n}ax}=\frac{1}{a(n-1){\cos }^{n-1}ax}+\int \frac{dx}{\sin ax{\cos }^{n-2}ax}\text{(for }n\ne 1\text{)}$

$\int \frac{dx}{{\sin }^{n}ax\cos ax}=-\frac{1}{a(n-1){\sin }^{n-1}ax}+\int \frac{dx}{{\sin }^{n-2}ax\cos ax}\text{(for }n\ne 1\text{)}$

$\int \frac{\sin axdx}{{\cos }^{n}ax}=\frac{1}{a(n-1){\cos }^{n-1}ax}+C\text{(for }n\ne 1\text{)}$

$\int \frac{{\sin }^{2}axdx}{\cos ax}=-\frac{1}{a}\sin ax+\frac{1}{a}\ln |\tan (\frac{\pi }{4}+\frac{ax}{2})|+C$

$\int \frac{{\sin }^{2}axdx}{{\cos }^{n}ax}=\frac{\sin ax}{a(n-1){\cos }^{n-1}ax}-\frac{1}{n-1}\int \frac{dx}{{\cos }^{n-2}ax}\text{(for }n\ne 1\text{)}$

$\int \frac{{\sin }^{n}axdx}{\cos ax}=-\frac{{\sin }^{n-1}ax}{a(n-1)}+\int \frac{{\sin }^{n-2}axdx}{\cos ax}\text{(for }n\ne 1\text{)}$

$\int \frac{{\sin }^{n}axdx}{{\cos }^{m}ax}=\frac{{\sin }^{n+1}ax}{a(m-1){\cos }^{m-1}ax}-\frac{n-m+2}{m-1}\int \frac{{\sin }^{n}axdx}{{\cos }^{m-2}ax}\text{(for }m\ne 1\text{)}$

also: $\int \frac{{\sin }^{n}axdx}{{\cos }^{m}ax}=-\frac{{\sin }^{n-1}ax}{a(n-m){\cos }^{m-1}ax}+\frac{n-1}{n-m}\int \frac{{\sin }^{n-2}axdx}{{\cos }^{m}ax}\text{(for }m\ne n\text{)}$

also: $\int \frac{{\sin }^{n}axdx}{{\cos }^{m}ax}=\frac{{\sin }^{n-1}ax}{a(m-1){\cos }^{m-1}ax}-\frac{n-1}{m-1}\int \frac{{\sin }^{n-2}axdx}{{\cos }^{m-2}ax}\text{(for }m\ne 1\text{)}$

$\int \frac{\cos axdx}{{\sin }^{n}ax}=-\frac{1}{a(n-1){\sin }^{n-1}ax}+C\text{(for }n\ne 1\text{)}$

$\int \frac{{\cos }^{2}axdx}{\sin ax}=\frac{1}{a}(\cos ax+\ln |\tan \frac{ax}{2}|)+C$

$\int \frac{{\cos }^{2}axdx}{{\sin }^{n}ax}=-\frac{1}{n-1}(\frac{\cos ax}{a{\sin }^{n-1}ax)}+\int \frac{dx}{{\sin }^{n-2}ax})\text{(for }n\ne 1\text{)}$

$\int \frac{{\cos }^{n}axdx}{{\sin }^{m}ax}=-\frac{{\cos }^{n+1}ax}{a(m-1){\sin }^{m-1}ax}-\frac{n-m-2}{m-1}\int \frac{{\cos }^{n}axdx}{{\sin }^{m-2}ax}\text{(for }m\ne 1\text{)}$

also: $\int \frac{{\cos }^{n}axdx}{{\sin }^{m}ax}=\frac{{\cos }^{n-1}ax}{a(n-m){\sin }^{m-1}ax}+\frac{n-1}{n-m}\int \frac{{\cos }^{n-2}axdx}{{\sin }^{m}ax}\text{(for }m\ne n\text{)}$

also: $\int \frac{{\cos }^{n}axdx}{{\sin }^{m}ax}=-\frac{{\cos }^{n-1}ax}{a(m-1){\sin }^{m-1}ax}-\frac{n-1}{m-1}\int \frac{{\cos }^{n-2}axdx}{{\sin }^{m-2}ax}\text{(for }m\ne 1\text{)}$

$\int \sin ax\tan axdx=\frac{1}{a}(\ln |\sec ax+\tan ax|-\sin ax)+C$

$\int \frac{{\tan }^{n}axdx}{{\sin }^{2}ax}=\frac{1}{a(n-1)}{\tan }^{n-1}(ax)+C\text{(for }n\ne 1\text{)}$

$\int \frac{{\tan }^{n}axdx}{{\cos }^{2}ax}=\frac{1}{a(n+1)}{\tan }^{n+1}ax+C\text{(for }n\ne -1\text{)}$

$\int \frac{{\cot }^{n}axdx}{{\sin }^{2}ax}=\frac{1}{a(n+1)}{\cot }^{n+1}ax+C\text{(for }n\ne -1\text{)}$

$\int \frac{{\cot }^{n}axdx}{{\cos }^{2}ax}=\frac{1}{a(1-n)}{\tan }^{1-n}ax+C\text{(for }n\ne 1\text{)}$

$\underset{-c}{\overset{c}{\int }}\sin xdx=0$

$\underset{-c}{\overset{c}{\int }}\cos xdx=2\underset{0}{\overset{c}{\int }}\cos xdx=2\underset{-c}{\overset{0}{\int }}\cos xdx=2\sin c$

$\underset{-c}{\overset{c}{\int }}\tan xdx=0$

$\underset{\frac{-a}{2}}{\overset{\frac{a}{2}}{\int }}{x}^2{\cos }^2\frac{n\pi x}{a}dx=\frac{a^3(n^2{\pi }^2-6)}{24n^2{\pi }^2}\text{(for }n=1,3,5...\text{)}$

$\int \arcsin \frac{x}{c}dx=x\arcsin \frac{x}{c}+\sqrt{c^2-x^2}$

$\int x\arcsin \frac{x}{c}dx=(\frac{x^2}{2}-\frac{c^2}{4})\arcsin \frac{x}{c}+\frac{x}{4}\sqrt{c^2-x^2}$

$\int x^2\arcsin \frac{x}{c}dx=\frac{x^3}{3}\arcsin \frac{x}{c}+\frac{x^2+2c^2}{9}\sqrt{c^2-x^2}$

$\int x^n{\sin }^{-1}xdx=\frac{1}{n+1}(x^{n+1}{\sin }^{-1}x$

$+\frac{x^n\sqrt{1-x^2}-n{x}^{n-1}{\sin }^{-1}x}{n-1}+n\int x^{n-2}{\sin }^{-1}xdx)$

$\int \arccos \frac{x}{c}dx=x\arccos \frac{x}{c}-\sqrt{c^2-x^2}$

$\int x\arccos \frac{x}{c}dx=(\frac{x^2}{2}-\frac{c^2}{4})\arccos \frac{x}{c}-\frac{x}{4}\sqrt{c^2-x^2}$

$\int x^2\arccos \frac{x}{c}dx=\frac{x^3}{3}\arccos \frac{x}{c}-\frac{x^2+2c^2}{9}\sqrt{c^2-x^2}$

$\int \arctan \frac{x}{c}dx=x\arctan \frac{x}{c}-\frac{c}{2}\ln (c^2+x^2)$

$\int x\arctan \frac{x}{c}dx=\frac{c^2+x^2}{2}\arctan \frac{x}{c}-\frac{cx}{2}$

$\int x^2\arctan \frac{x}{c}dx=\frac{x^3}{3}\arctan \frac{x}{c}-\frac{c{x}^2}{6}+\frac{c^3}{6}\ln c^2+x^2$

$\int x^n\arctan \frac{x}{c}dx=\frac{x^{n+1}}{n+1}\arctan \frac{x}{c}-\frac{c}{n+1}\int \frac{x^{n+1}dx}{c^2+x^2}\text{( }n\ne 1\text{)}$

$\int \mathrm{arcsec}\frac{x}{c}dx=x\mathrm{arcsec}\frac{x}{c}+\frac{x}{c|x|}\ln |x\pm \sqrt{x^2-1}|$

$\int x\mathrm{arcsec}xdx=\frac{1}{2}(x^2\mathrm{arcsec}x-\sqrt{x^2-1})$

$\int x^n\mathrm{arcsec}xdx=\frac{1}{n+1}(x^{n+1}\mathrm{arcsec}x-\frac{1}{n}(x^{n-1}\sqrt{x^2-1}$

$+(1-n)(x^{n-1}\mathrm{arcsec}x+(1-n)\int x^{n-2}\mathrm{arcsec}xdx)))$

$\int \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\frac{x}{c}dx=x\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\frac{x}{c}+\frac{c}{2}\ln (c^2+x^2)$

$\int x\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\frac{x}{c}dx=\frac{c^2+x^2}{2}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\frac{x}{c}+\frac{cx}{2}$

$\int x^2\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\frac{x}{c}dx=\frac{x^3}{3}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\frac{x}{c}+\frac{c{x}^2}{6}-\frac{c^3}{6}\ln (c^2+x^2)$

$\int x^n\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\frac{x}{c}dx=\frac{x^{n+1}}{n+1}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\frac{x}{c}+\frac{c}{n+1}\int \frac{x^{n+1}dx}{c^2+x^2}\text{( }n\ne 1\text{)}$

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

$\int \frac{dx}{ax+b}=\frac{1}{a}\ln |ax+b|$

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

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

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

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

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

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

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

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

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

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

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

$\int \frac{dx}{x^2+a^2}=\frac{1}{a}\arctan \frac{x}{a}$

$\int \frac{dx}{x^2-a^2}=-\frac{1}{a}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\frac{x}{a}=\frac{1}{2a}\ln \frac{a-x}{a+x}\text{( }|x|<|a|\text{)}$

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

$\int \frac{dx}{a{x}^2+bx+c}=\frac{2}{\sqrt{4ac-b^2}}\arctan \frac{2ax+b}{\sqrt{4ac-b^2}}\text{( }4ac-b^2>0\text{)}$

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

$\int \frac{dx}{a{x}^2+bx+c}=-\frac{2}{2ax+b}\text{( }4ac-b^2=0\text{)}$

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

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

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

$\int \frac{mx+n}{a{x}^2+bx+c}dx=\frac{m}{2a}\ln |a{x}^2+bx+c|-\frac{2an-bm}{a(2ax+b)}\text{( }4ac-b^2=0\text{)}$

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

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

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

$\int rdx=\frac{1}{2}(xr+a^2\ln (x+r))$

$\int r^{3}dx=\frac{1}{4}x{r}^3+\frac{3}{8}{a}^{2}xr+\frac{3}{8}{a}^4\ln (x+r)$

$\int r^{5}dx=\frac{1}{6}x{r}^5+\frac{5}{24}{a}^{2}x{r}^3+\frac{5}{16}{a}^{4}xr+\frac{5}{16}{a}^6\ln (x+r)$

$\int xrdx=\frac{r^3}{3}$

$\int x{r}^{3}dx=\frac{r^5}{5}$

$\int x{r}^{2n+1}dx=\frac{r^{2n+3}}{2n+3}$

$\int x^{2}rdx=\frac{x{r}^3}{4}-\frac{a^{2}xr}{8}-\frac{a^4}{8}\ln (x+r)$

$\int x^2{r}^{3}dx=\frac{x{r}^5}{6}-\frac{a^{2}x{r}^3}{24}-\frac{a^{4}xr}{16}-\frac{a^6}{16}\ln (x+r)$

$\int x^{3}rdx=\frac{r^5}{5}-\frac{a^2{r}^3}{3}$

$\int x^3{r}^{3}dx=\frac{r^7}{7}-\frac{a^2{r}^5}{5}$

$\int x^3{r}^{2n+1}dx=\frac{r^{2n+5}}{2n+5}-\frac{a^3{r}^{2n+3}}{2n+3}$

$\int x^{4}rdx=\frac{x^3{r}^3}{6}-\frac{a^{2}x{r}^3}{8}+\frac{a^{4}xr}{16}+\frac{a^6}{16}\ln (x+r)$

$\int x^4{r}^{3}dx=\frac{x^3{r}^5}{8}-\frac{a^{2}x{r}^5}{16}+\frac{a^{4}x{r}^3}{64}+\frac{3a^{6}xr}{128}+\frac{3a^8}{128}\ln (x+r)$

$\int x^{5}rdx=\frac{r^7}{7}-\frac{2a^2{r}^5}{5}+\frac{a^4{r}^3}{3}$

$\int x^5{r}^{3}dx=\frac{r^9}{9}-\frac{2a^2{r}^7}{7}+\frac{a^4{r}^5}{5}$

$\int x^5{r}^{2n+1}dx=\frac{r^{2n+7}}{2n+7}-\frac{2a^2{r}^{2n+5}}{2n+5}+\frac{a^4{r}^{2n+3}}{2n+3}$

$\int \frac{rdx}{x}=r-a\ln \left|\frac{a+r}{x}\right|=r-a\mathrm{arsinh}\frac{a}{x}$

$\int \frac{r^{3}dx}{x}=\frac{r^3}{3}+a^{2}r-a^3\ln \left|\frac{a+r}{x}\right|$

$\int \frac{r^{5}dx}{x}=\frac{r^5}{5}+\frac{a^2{r}^3}{3}+a^{4}r-a^5\ln \left|\frac{a+r}{x}\right|$

$\int \frac{r^{7}dx}{x}=\frac{r^7}{7}+\frac{a^2{r}^5}{5}+\frac{a^4{r}^3}{3}+a^{6}r-a^7\ln \left|\frac{a+r}{x}\right|$

$\int \frac{dx}{r}=\mathrm{arsinh}\frac{x}{a}=\ln \left(\frac{x+r}{a}\right)$

$\int \frac{dx}{r^3}=\frac{x}{a^{2}r}$

$\int \frac{xdx}{r}=r$

$\int \frac{xdx}{r^3}=-\frac{1}{r}$

$\int \frac{x^{2}dx}{r}=\frac{x}{2}r-\frac{a^2}{2}\mathrm{arsinh}\frac{x}{a}=\frac{x}{2}r-\frac{a^2}{2}\ln \left(\frac{x+r}{a}\right)$

$\int \frac{dx}{xr}=-\frac{1}{a}\mathrm{arsinh}\frac{a}{x}=-\frac{1}{a}\ln \left|\frac{a+r}{x}\right|$