View source for Tích phân hàm số toán lủy thừa của cosecant

[[Category:Tích phân]]
[[Category:VI]]
:<math>\int \csc{ax} \, dx = -\frac{1}{a}\ln{\left| \csc{ax}+\cot{ax}\right|}+C</math>

:<math>\int \csc^2{x} \, dx = -\cot{x}+C</math>

:<math>\int \csc^n{ax} \, dx = -\frac{\csc^{n-1}{ax} \csc{ax}}{a(n-1)} \,+\, \frac{n-2}{n-1}\int \csc^{n-2}{ax} \, dx \qquad \mbox{ (for }n \ne 1\mbox{)}\,</math>

:<math>\int \frac{dx}{\csc{x} + 1} = x - \frac{2\sin{\frac{x}{2}}}{\cos{\frac{x}{2}}+\sin{\frac{x}{2}}}+C</math>

:<math>\int \frac{dx}{\csc{x} - 1} = \frac{2\sin{\frac{x}{2}}}{\cos{\frac{x}{2}}-\sin{\frac{x}{2}}}-x+C</math>


:<math>\int\cot ax\;dx = \frac{1}{a}\ln|\sin ax|+C\,\!</math>

: <math>\int\cot^n ax\;dx = -\frac{1}{a(n-1)}\cot^{n-1} ax - \int\cot^{n-2} ax\;dx \qquad\mbox{(for }n\neq 1\mbox{)}\,\!</math>

: <math>\int\frac{dx}{1 + \cot ax} = \int\frac{\tan ax\;dx}{\tan ax+1}\,\!</math>

: <math>\int\frac{dx}{1 - \cot ax} = \int\frac{\tan ax\;dx}{\tan ax-1}\,\!</math>



: <math>\int\frac{dx}{\cos ax\pm\sin ax} = \frac{1}{a\sqrt{2}}\ln\left|\tan\left(\frac{ax}{2}\pm\frac{\pi}{8}\right)\right|+C</math>

: <math>\int\frac{dx}{(\cos ax\pm\sin ax)^2} = \frac{1}{2a}\tan\left(ax\mp\frac{\pi}{4}\right)+C</math>

: <math>\int\frac{dx}{(\cos x + \sin x)^n} = \frac{1}{n-1}\left(\frac{\sin x - \cos x}{(\cos x + \sin x)^{n - 1}} - 2(n - 2)\int\frac{dx}{(\cos x + \sin x)^{n-2}} \right)</math>

: <math>\int\frac{\cos ax\;dx}{\cos ax + \sin ax} = \frac{x}{2} + \frac{1}{2a}\ln\left|\sin ax + \cos ax\right|+C</math>

: <math>\int\frac{\cos ax\;dx}{\cos ax - \sin ax} = \frac{x}{2} - \frac{1}{2a}\ln\left|\sin ax - \cos ax\right|+C</math>

: <math>\int\frac{\sin ax\;dx}{\cos ax + \sin ax} = \frac{x}{2} - \frac{1}{2a}\ln\left|\sin ax + \cos ax\right|+C</math>

: <math>\int\frac{\sin ax\;dx}{\cos ax - \sin ax} = -\frac{x}{2} - \frac{1}{2a}\ln\left|\sin ax - \cos ax\right|+C</math>

: <math>\int\frac{\cos ax\;dx}{\sin ax(1+\cos ax)} = -\frac{1}{4a}\tan^2\frac{ax}{2}+\frac{1}{2a}\ln\left|\tan\frac{ax}{2}\right|+C</math>

: <math>\int\frac{\cos ax\;dx}{\sin ax(1-\cos ax)} = -\frac{1}{4a}\cot^2\frac{ax}{2}-\frac{1}{2a}\ln\left|\tan\frac{ax}{2}\right|+C</math>

: <math>\int\frac{\sin ax\;dx}{\cos ax(1+\sin ax)} = \frac{1}{4a}\cot^2\left(\frac{ax}{2}+\frac{\pi}{4}\right)+\frac{1}{2a}\ln\left|\tan\left(\frac{ax}{2}+\frac{\pi}{4}\right)\right|+C</math>

: <math>\int\frac{\sin ax\;dx}{\cos ax(1-\sin ax)} = \frac{1}{4a}\tan^2\left(\frac{ax}{2}+\frac{\pi}{4}\right)-\frac{1}{2a}\ln\left|\tan\left(\frac{ax}{2}+\frac{\pi}{4}\right)\right|+C</math>

: <math>\int\sin ax\cos ax\;dx = -\frac{1}{2a}\cos^2 ax +C\,\!</math>

: <math>\int\sin a_1x\cos a_2x\;dx = -\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\qquad\mbox{(for }|a_1|\neq|a_2|\mbox{)}\,\!</math>

: <math>\int\sin^n ax\cos ax\;dx = \frac{1}{a(n+1)}\sin^{n+1} ax +C\qquad\mbox{(for }n\neq -1\mbox{)}\,\!</math>

: <math>\int\sin ax\cos^n ax\;dx = -\frac{1}{a(n+1)}\cos^{n+1} ax +C\qquad\mbox{(for }n\neq -1\mbox{)}\,\!</math>

: <math>\int\sin^n ax\cos^m ax\;dx = -\frac{\sin^{n-1} ax\cos^{m+1} ax}{a(n+m)}+\frac{n-1}{n+m}\int\sin^{n-2} ax\cos^m ax\;dx  \qquad\mbox{(for }m,n>0\mbox{)}\,\!</math>

: và: <math>\int\sin^n ax\cos^m ax\;dx = \frac{\sin^{n+1} ax\cos^{m-1} ax}{a(n+m)} + \frac{m-1}{n+m}\int\sin^n ax\cos^{m-2} ax\;dx \qquad\mbox{(for }m,n>0\mbox{)}\,\!</math>

: <math>\int\frac{dx}{\sin ax\cos ax} = \frac{1}{a}\ln\left|\tan ax\right|+C</math>

: <math>\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} \qquad\mbox{(for }n\neq 1\mbox{)}\,\!</math>

: <math>\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} \qquad\mbox{(for }n\neq 1\mbox{)}\,\!</math>

: <math>\int\frac{\sin ax\;dx}{\cos^n ax} = \frac{1}{a(n-1)\cos^{n-1} ax} +C\qquad\mbox{(for }n\neq 1\mbox{)}\,\!</math>

: <math>\int\frac{\sin^2 ax\;dx}{\cos ax} = -\frac{1}{a}\sin ax+\frac{1}{a}\ln\left|\tan\left(\frac{\pi}{4}+\frac{ax}{2}\right)\right|+C</math>

: <math>\int\frac{\sin^2 ax\;dx}{\cos^n ax} = \frac{\sin ax}{a(n-1)\cos^{n-1}ax}-\frac{1}{n-1}\int\frac{dx}{\cos^{n-2}ax} \qquad\mbox{(for }n\neq 1\mbox{)}\,\!</math>

: <math>\int\frac{\sin^n ax\;dx}{\cos ax} = -\frac{\sin^{n-1} ax}{a(n-1)} + \int\frac{\sin^{n-2} ax\;dx}{\cos ax} \qquad\mbox{(for }n\neq 1\mbox{)}\,\!</math>

: <math>\int\frac{\sin^n ax\;dx}{\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 ax\;dx}{\cos^{m-2} ax} \qquad\mbox{(for }m\neq 1\mbox{)}\,\!</math>

: và: <math>\int\frac{\sin^n ax\;dx}{\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} ax\;dx}{\cos^m ax} \qquad\mbox{(for }m\neq n\mbox{)}\,\!</math>

: và: <math>\int\frac{\sin^n ax\;dx}{\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} ax\;dx}{\cos^{m-2} ax} \qquad\mbox{(for }m\neq 1\mbox{)}\,\!</math>

: <math>\int\frac{\cos ax\;dx}{\sin^n ax} = -\frac{1}{a(n-1)\sin^{n-1} ax} +C\qquad\mbox{(for }n\neq 1\mbox{)}\,\!</math>

: <math>\int\frac{\cos^2 ax\;dx}{\sin ax} = \frac{1}{a}\left(\cos ax+\ln\left|\tan\frac{ax}{2}\right|\right) +C</math>

: <math>\int\frac{\cos^2 ax\;dx}{\sin^n ax} = -\frac{1}{n-1}\left(\frac{\cos ax}{a\sin^{n-1} ax)}+\int\frac{dx}{\sin^{n-2} ax}\right) \qquad\mbox{(for }n\neq 1\mbox{)}</math>

: <math>\int\frac{\cos^n ax\;dx}{\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 ax\;dx}{\sin^{m-2} ax} \qquad\mbox{(for }m\neq 1\mbox{)}\,\!</math>

: và: <math>\int\frac{\cos^n ax\;dx}{\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} ax\;dx}{\sin^m ax} \qquad\mbox{(for }m\neq n\mbox{)}\,\!</math>

: và: <math>\int\frac{\cos^n ax\;dx}{\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} ax\;dx}{\sin^{m-2} ax} \qquad\mbox{(for }m\neq 1\mbox{)}\,\!</math>



: <math>\int \sin ax \tan ax\;dx = \frac{1}{a}(\ln|\sec ax + \tan ax| - \sin ax)+C\,\!</math>

: <math>\int\frac{\tan^n ax\;dx}{\sin^2 ax} = \frac{1}{a(n-1)}\tan^{n-1} (ax) +C\qquad\mbox{(for }n\neq 1\mbox{)}\,\!</math>



: <math>\int\frac{\tan^n ax\;dx}{\cos^2 ax} = \frac{1}{a(n+1)}\tan^{n+1} ax +C\qquad\mbox{(for }n\neq -1\mbox{)}\,\!</math>

: <math>\int\frac{\cot^n ax\;dx}{\sin^2 ax} = -\frac{1}{a(n+1)}\cot^{n+1} ax  +C\qquad\mbox{(for }n\neq -1\mbox{)}\,\!</math>



: <math>\int\frac{\cot^n ax\;dx}{\cos^2 ax} = \frac{1}{a(1-n)}\tan^{1-n} ax +C\qquad\mbox{(for }n\neq 1\mbox{)}\,\!</math>



: <math>\int\arcsin\frac{x}{c}\,dx = x\arcsin\frac{x}{c} + \sqrt{c^2-x^2}</math>

: <math>\int x \arcsin\frac{x}{c}\,dx = \left(\frac{x^2}{2}-\frac{c^2}{4}\right)\arcsin\frac{x}{c} + \frac{x}{4}\sqrt{c^2-x^2}</math>

: <math>\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}</math>

: <math>\int x^n \sin^{-1}x\,dx = \frac{1}{n+1}\left(x^{n+1}\sin^{-1}x  \right.</math>
:::<math>\left. + \frac{x^n\sqrt{1 - x^2} - nx^{n-1}\sin^{-1}x}{n-1} + n\int x^{n-2}\sin^{-1}x\,dx\right)</math>

: <math>\int\arccos\frac{x}{c}\,dx = x\arccos\frac{x}{c} - \sqrt{c^2-x^2}</math>

: <math>\int x \arccos\frac{x}{c}\,dx = \left(\frac{x^2}{2}-\frac{c^2}{4}\right)\arccos\frac{x}{c} - \frac{x}{4}\sqrt{c^2-x^2}</math>

: <math>\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}</math>

: <math>\int\arctan\frac{x}{c}\,dx = x\arctan\frac{x}{c} - \frac{c}{2}\ln(c^2+x^2)</math>

: <math>\int x \arctan\frac{x}{c}\,dx = \frac{c^2+x^2}{2}\arctan\frac{x}{c} - \frac{cx}{2}</math>

: <math>\int x^2 \arctan\frac{x}{c}\,dx = \frac{x^3}{3}\arctan\frac{x}{c} - \frac{cx^2}{6} + \frac{c^3}{6}\ln{c^2+x^2}</math>

: <math>\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} \qquad\mbox{(}n\neq 1\mbox{)}</math>

: <math>\int \arcsec{\frac{x}{c}}\,dx = x \arcsec{\frac{x}{c}} + \frac{x}{c|x|}\ln{|x \pm \sqrt{x^2 - 1}|}</math>

: <math>\int x\arcsec{x}\,dx\,=\,\frac{1}{2}\left(x^2\arcsec{x} - \sqrt{x^2 - 1}\right) </math>

: <math>\int x^n\arcsec{x}\,dx\,=\,\frac{1}{n+1}\left(x^{n+1}\arcsec{x} - \frac{1}{n}\left(x^{n-1}\sqrt{x^2 - 1}\; \right. \right.</math>
::: <math>\left. \left. + (1-n)\left(x^{n-1}\arcsec{x} + (1-n)\int x^{n-2}\arcsec{x}\,dx \right)\right)\right)</math>

: <math>\int\mathrm{arccot}\,\frac{x}{c}\,dx = x\,\mathrm{arccot}\,\frac{x}{c} + \frac{c}{2}\ln(c^2+x^2)</math>

: <math>\int x\,\mathrm{arccot}\,\frac{x}{c}\,dx = \frac{c^2+x^2}{2}\,\mathrm{arccot}\,\frac{x}{c} + \frac{cx}{2}</math>

: <math>\int x^2\,\mathrm{arccot}\,\frac{x}{c}\,dx = \frac{x^3}{3}\,\mathrm{arccot}\,\frac{x}{c} + \frac{cx^2}{6} - \frac{c^3}{6}\ln(c^2+x^2)</math>

: <math>\int x^n\,\mathrm{arccot}\,\frac{x}{c}\,dx = \frac{x^{n+1}}{n+1}\,\mathrm{arccot}\,\frac{x}{c} + \frac{c}{n+1}\int\frac{x^{n+1} dx}{c^2+x^2} \qquad\mbox{(}n\neq 1\mbox{)}</math>


: <math>\int\sinh cx\,dx = \frac{1}{c}\cosh cx</math>

: <math>\int\cosh cx\,dx = \frac{1}{c}\sinh cx</math>

: <math>\int\sinh^2 cx\,dx = \frac{1}{4c}\sinh 2cx - \frac{x}{2}</math>

: <math>\int\cosh^2 cx\,dx = \frac{1}{4c}\sinh 2cx + \frac{x}{2}</math>

: <math>\int\sinh^n cx\,dx = \frac{1}{cn}\sinh^{n-1} cx\cosh cx - \frac{n-1}{n}\int\sinh^{n-2} cx\,dx \qquad\mbox{(}n>0\mbox{)}</math>

: hay: <math>\int\sinh^n cx\,dx = \frac{1}{c(n+1)}\sinh^{n+1} cx\cosh cx - \frac{n+2}{n+1}\int\sinh^{n+2}cx\,dx \qquad\mbox{(}n<0\mbox{, }n\neq -1\mbox{)}</math>

: <math>\int\cosh^n cx\,dx = \frac{1}{cn}\sinh cx\cosh^{n-1} cx + \frac{n-1}{n}\int\cosh^{n-2} cx\,dx \qquad\mbox{(}n>0\mbox{)}</math>

: hay: <math>\int\cosh^n cx\,dx = -\frac{1}{c(n+1)}\sinh cx\cosh^{n+1} cx - \frac{n+2}{n+1}\int\cosh^{n+2}cx\,dx \qquad\mbox{(}n<0\mbox{, }n\neq -1\mbox{)}</math>

: <math>\int\frac{dx}{\sinh cx} = \frac{1}{c} \ln\left|\tanh\frac{cx}{2}\right|</math>

: hay: <math>\int\frac{dx}{\sinh cx} = \frac{1}{c} \ln\left|\frac{\cosh cx - 1}{\sinh cx}\right|</math>

: hay: <math>\int\frac{dx}{\sinh cx} = \frac{1}{c} \ln\left|\frac{\sinh cx}{\cosh cx + 1}\right|</math>

: hay: <math>\int\frac{dx}{\sinh cx} = \frac{1}{c} \ln\left|\frac{\cosh cx - 1}{\cosh cx + 1}\right|</math>

: <math>\int\frac{dx}{\cosh cx} = \frac{2}{c} \arctan e^{cx}</math>

: <math>\int\frac{dx}{\sinh^n cx} = \frac{\cosh cx}{c(n-1)\sinh^{n-1} cx}-\frac{n-2}{n-1}\int\frac{dx}{\sinh^{n-2} cx} \qquad\mbox{(}n\neq 1\mbox{)}</math>

: <math>\int\frac{dx}{\cosh^n cx} = \frac{\sinh cx}{c(n-1)\cosh^{n-1} cx}+\frac{n-2}{n-1}\int\frac{dx}{\cosh^{n-2} cx} \qquad\mbox{(}n\neq 1\mbox{)}</math>

: <math>\int\frac{\cosh^n cx}{\sinh^m cx} dx = \frac{\cosh^{n-1} cx}{c(n-m)\sinh^{m-1} cx} + \frac{n-1}{n-m}\int\frac{\cosh^{n-2} cx}{\sinh^m cx} dx \qquad\mbox{(}m\neq n\mbox{)}</math>

: hay: <math>\int\frac{\cosh^n cx}{\sinh^m cx} dx = -\frac{\cosh^{n+1} cx}{c(m-1)\sinh^{m-1} cx} + \frac{n-m+2}{m-1}\int\frac{\cosh^n cx}{\sinh^{m-2} cx} dx \qquad\mbox{(}m\neq 1\mbox{)}</math>

: hay: <math>\int\frac{\cosh^n cx}{\sinh^m cx} dx = -\frac{\cosh^{n-1} cx}{c(m-1)\sinh^{m-1} cx} + \frac{n-1}{m-1}\int\frac{\cosh^{n-2} cx}{\sinh^{m-2} cx} dx \qquad\mbox{(}m\neq 1\mbox{)}</math>

: <math>\int\frac{\sinh^m cx}{\cosh^n cx} dx = \frac{\sinh^{m-1} cx}{c(m-n)\cosh^{n-1} cx} + \frac{m-1}{m-n}\int\frac{\sinh^{m-2} cx}{\cosh^n cx} dx \qquad\mbox{(}m\neq n\mbox{)}</math>

: hay: <math>\int\frac{\sinh^m cx}{\cosh^n cx} dx = \frac{\sinh^{m+1} cx}{c(n-1)\cosh^{n-1} cx} + \frac{m-n+2}{n-1}\int\frac{\sinh^m cx}{\cosh^{n-2} cx} dx \qquad\mbox{(}n\neq 1\mbox{)}</math>

: hay: <math>\int\frac{\sinh^m cx}{\cosh^n cx} dx = -\frac{\sinh^{m-1} cx}{c(n-1)\cosh^{n-1} cx} + \frac{m-1}{n-1}\int\frac{\sinh^{m -2} cx}{\cosh^{n-2} cx} dx \qquad\mbox{(}n\neq 1\mbox{)}</math>

: <math>\int x\sinh cx\,dx = \frac{1}{c} x\cosh cx - \frac{1}{c^2}\sinh cx</math>

: <math>\int x\cosh cx\,dx = \frac{1}{c} x\sinh cx - \frac{1}{c^2}\cosh cx</math>

: <math>\int \tanh cx\,dx = \frac{1}{c}\ln|\cosh cx|</math>

: <math>\int \coth cx\,dx = \frac{1}{c}\ln|\sinh cx|</math>

: <math>\int \tanh^n cx\,dx = -\frac{1}{c(n-1)}\tanh^{n-1} cx+\int\tanh^{n-2} cx\,dx \qquad\mbox{(}n\neq 1\mbox{)}</math>

: <math>\int \coth^n cx\,dx = -\frac{1}{c(n-1)}\coth^{n-1} cx+\int\coth^{n-2} cx\,dx \qquad\mbox{(}n\neq 1\mbox{)}</math>

: <math>\int \sinh bx \sinh cx\,dx = \frac{1}{b^2-c^2} (b\sinh cx \cosh bx - c\cosh cx \sinh bx) \qquad\mbox{(}b^2\neq c^2\mbox{)}</math>

: <math>\int \cosh bx \cosh cx\,dx = \frac{1}{b^2-c^2} (b\sinh bx \cosh cx - c\sinh cx \cosh bx) \qquad\mbox{(}b^2\neq c^2\mbox{)}</math>

: <math>\int \cosh bx \sinh cx\,dx = \frac{1}{b^2-c^2} (b\sinh bx \sinh cx - c\cosh bx \cosh cx) \qquad\mbox{(}b^2\neq c^2\mbox{)}</math>

: <math>\int \sinh (ax+b)\sin (cx+d)\,dx = \frac{a}{a^2+c^2}\cosh(ax+b)\sin(cx+d)-\frac{c}{a^2+c^2}\sinh(ax+b)\cos(cx+d)</math>

: <math>\int \sinh (ax+b)\cos (cx+d)\,dx = \frac{a}{a^2+c^2}\cosh(ax+b)\cos(cx+d)+\frac{c}{a^2+c^2}\sinh(ax+b)\sin(cx+d)</math>

: <math>\int \cosh (ax+b)\sin (cx+d)\,dx = \frac{a}{a^2+c^2}\sinh(ax+b)\sin(cx+d)-\frac{c}{a^2+c^2}\cosh(ax+b)\cos(cx+d)</math>

: <math>\int \cosh (ax+b)\cos (cx+d)\,dx = \frac{a}{a^2+c^2}\sinh(ax+b)\cos(cx+d)+\frac{c}{a^2+c^2}\cosh(ax+b)\sin(cx+d)</math>



: <math>\int\mathrm{arsinh}\,\frac{x}{c}\,dx = x\,\mathrm{arsinh}\,\frac{x}{c}-\sqrt{x^2+c^2}</math>

: <math>\int\mathrm{arcosh}\,\frac{x}{c}\,dx = x\,\mathrm{arcosh}\,\frac{x}{c}-\sqrt{x^2-c^2}</math>

: <math>\int\mathrm{artanh}\,\frac{x}{c}\,dx = x\,\mathrm{artanh}\,\frac{x}{c} + \frac{c}{2}\ln|c^2 - x^2| \qquad\mbox{(}|x|<|c|\mbox{)}</math>

: <math>\int\mathrm{arcoth}\,\frac{x}{c}\,dx = x\,\mathrm{arcoth}\,\frac{x}{c} + \frac{c}{2}\ln|x^2 - c^2| \qquad\mbox{(}|x|>|c|\mbox{)}</math>

: <math>\int\mathrm{arsech}\,\frac{x}{c}\,dx = x\,\mathrm{arsech}\,\frac{x}{c} - c\,\mathrm{arctan}\,\frac{x\,\sqrt{\frac{c - x}{c + x}}}{x - c} \qquad\mbox{(} x \in (0,\,c) \mbox{)}</math>

: <math>\int\mathrm{arcsch}\,\frac{x}{c}\,dx = x\,\mathrm{arcsch}\,\frac{x}{c} + c\,\ln\,\frac{x + \sqrt{x^2 + c^2}}{c} \qquad\mbox{(} x \in (0,\,c) \mbox{)}</math>

: <math>\int e^{cx}\;dx = \frac{1}{c} e^{cx}</math>

: <math>\int a^{cx}\;dx = \frac{1}{c \ln a} a^{cx} \qquad\mbox{(} a > 0,\mbox{ }a \ne 1\mbox{)}</math>

: <math>\int xe^{cx}\; dx = \frac{e^{cx}}{c^2}(cx-1)</math>

: <math>\int x^2 e^{cx}\;dx = e^{cx}\left(\frac{x^2}{c}-\frac{2x}{c^2}+\frac{2}{c^3}\right)</math>

: <math>\int x^n e^{cx}\; dx = \frac{1}{c} x^n e^{cx} - \frac{n}{c}\int x^{n-1} e^{cx} dx</math>

: <math>\int\frac{e^{cx}\; dx}{x} = \ln|x| +\sum_{i=1}^\infty\frac{(cx)^i}{i\cdot i!}</math>

: <math>\int\frac{e^{cx}\; dx}{x^n} = \frac{1}{n-1}\left(-\frac{e^{cx}}{x^{n-1}}+c\int\frac{e^{cx} }{x^{n-1}}\,dx\right) \qquad\mbox{(}n\neq 1\mbox{)}</math>

: <math>\int e^{cx}\ln x\; dx = \frac{1}{c}e^{cx}\ln|x|-\operatorname{Ei}\,(cx)</math>

: <math>\int e^{cx}\sin bx\; dx = \frac{e^{cx}}{c^2+b^2}(c\sin bx - b\cos bx)</math>

: <math>\int e^{cx}\cos bx\; dx = \frac{e^{cx}}{c^2+b^2}(c\cos bx + b\sin bx)</math>

: <math>\int e^{cx}\sin^n x\; dx = \frac{e^{cx}\sin^{n-1} x}{c^2+n^2}(c\sin x-n\cos x)+\frac{n(n-1)}{c^2+n^2}\int e^{cx}\sin^{n-2} x\;dx</math>

: <math>\int e^{cx}\cos^n x\; dx = \frac{e^{cx}\cos^{n-1} x}{c^2+n^2}(c\cos x+n\sin x)+\frac{n(n-1)}{c^2+n^2}\int e^{cx}\cos^{n-2} x\;dx</math>

:<math>\int x e^{c x^2 }\; dx= \frac{1}{2c} \;  e^{c x^2}</math>

:<math>\int {1 \over \sigma\sqrt{2\pi} }\,e^{-{(x-\mu)^2 / 2\sigma^2}}\; dx= \frac{1}{2 \sigma} (1 + \mbox{erf}\,\frac{x-\mu}{\sigma \sqrt{2}})</math>

:<math>\int e^{x^2}\,dx = e^{x^2}\left(\sum_{j=0}^{n-1}c_{2j}\,\frac{1}{x^{2j+1}} \right)+(2n-1)c_{2n-2} \int \frac{e^{x^2}}{x^{2n}}\;dx  \quad (n > 0),  </math>
::với <math>c_{2j}=\frac{ 1 \cdot 3 \cdot 5 \cdots (2j-1)}{2^{j+1}}=\frac{(2j)\,!}{j!\, 2^{2j+1}} \ . </math>

:<math>\int_{-\infty}^{\infty} e^{-ax^2}\,dx=\sqrt{\pi \over a}</math>

:<math>\int_{0}^{\infty} x^{2n} e^{-{x^2}/{a^2}}\,dx=\sqrt{\pi} {(2n)! \over {n!}} {\left (\frac{a}{2} \right)}^{2n + 1}</math>

: <math>\int\ln cx\,dx = x\ln cx - x</math>

: <math>\int (\ln x)^2\; dx = x(\ln x)^2 - 2x\ln x + 2x</math>

: <math>\int (\ln cx)^n\; dx = x(\ln cx)^n - n\int (\ln cx)^{n-1} dx</math>

: <math>\int \frac{dx}{\ln x} = \ln|\ln x| + \ln x + \sum^\infty_{i=2}\frac{(\ln x)^i}{i\cdot i!}</math>

: <math>\int \frac{dx}{(\ln x)^n} = -\frac{x}{(n-1)(\ln x)^{n-1}} + \frac{1}{n-1}\int\frac{dx}{(\ln x)^{n-1}} \qquad\mbox{(}n\neq 1\mbox{)}</math>

: <math>\int x^m\ln x\;dx = x^{m+1}\left(\frac{\ln x}{m+1}-\frac{1}{(m+1)^2}\right) \qquad\mbox{(}m\neq -1\mbox{)}</math>

: <math>\int x^m (\ln x)^n\; dx = \frac{x^{m+1}(\ln x)^n}{m+1} - \frac{n}{m+1}\int x^m (\ln x)^{n-1} dx  \qquad\mbox{(}m\neq -1\mbox{)}</math>

: <math>\int \frac{(\ln x)^n\; dx}{x} = \frac{(\ln x)^{n+1}}{n+1}  \qquad\mbox{(}n\neq -1\mbox{)}</math>

: <math>\int \frac{\ln x\,dx}{x^m} = -\frac{\ln x}{(m-1)x^{m-1}}-\frac{1}{(m-1)^2 x^{m-1}} \qquad\mbox{(}m\neq 1\mbox{)}</math>

: <math>\int \frac{(\ln x)^n\; dx}{x^m} = -\frac{(\ln x)^n}{(m-1)x^{m-1}} + \frac{n}{m-1}\int\frac{(\ln x)^{n-1} dx}{x^m} \qquad\mbox{( }m\neq 1\mbox{)}</math>

: <math>\int \frac{x^m\; dx}{(\ln x)^n} = -\frac{x^{m+1}}{(n-1)(\ln x)^{n-1}} + \frac{m+1}{n-1}\int\frac{x^m dx}{(\ln x)^{n-1}}  \qquad\mbox{( }n\neq 1\mbox{)}</math>

: <math>\int \frac{dx}{x\ln x} = \ln|\ln x|</math>

: <math>\int \frac{dx}{x^n\ln x} = \ln|\ln x| + \sum^\infty_{i=1} (-1)^i\frac{(n-1)^i(\ln x)^i}{i\cdot i!}</math>

: <math>\int \frac{dx}{x (\ln x)^n} = -\frac{1}{(n-1)(\ln x)^{n-1}} \qquad\mbox{(}n\neq 1\mbox{)}</math>

: <math>\int \sin (\ln x)\;dx = \frac{x}{2}(\sin (\ln x) - \cos (\ln x))</math>

: <math>\int \cos (\ln x)\;dx = \frac{x}{2}(\sin (\ln x) + \cos (\ln x))</math>


: <math>\int (ax + b)^n dx = \frac{(ax + b)^{n+1}}{a(n + 1)} \qquad\mbox{(} n\neq -1\mbox{)}\,\!</math>

: <math>\int\frac{dx}{ax + b} = \frac{1}{a}\ln\left|ax + b\right|</math>

: <math>\int x(ax + b)^n dx = \frac{a(n + 1)x - b}{a^2(n + 1)(n + 2)} (ax + b)^{n+1} \qquad\mbox{(}n \not\in \{1, 2\}\mbox{)}</math>

: <math>\int\frac{x}{ax + b}dx = \frac{x}{a} - \frac{b}{a^2}\ln\left|ax + b\right|</math>

: <math>\int\frac{x}{(ax + b)^2}dx = \frac{b}{a^2(ax + b)} + \frac{1}{a^2}\ln\left|ax + b\right|</math>

: <math>\int\frac{x}{(ax + b)^n}dx = \frac{a(1 - n)x - b}{a^2(n - 1)(n - 2)(ax + b)^{n-1}} \qquad\mbox{(} n\not\in \{1, 2\}\mbox{)}</math>

: <math>\int\frac{x^2}{ax + b}dx = \frac{1}{a^3}\left(\frac{(ax + b)^2}{2} - 2b(ax + b) + b^2\ln\left|ax + b\right|\right)</math>

: <math>\int\frac{x^2}{(ax + b)^2}dx = \frac{1}{a^3}\left(ax + b - 2b\ln\left|ax + b\right| - \frac{b^2}{ax + b}\right)</math>

: <math>\int\frac{x^2}{(ax + b)^3}dx = \frac{1}{a^3}\left(\ln\left|ax + b\right| + \frac{2b}{ax + b} - \frac{b^2}{2(ax + b)^2}\right)</math>

: <math>\int\frac{x^2}{(ax + b)^n}dx = \frac{1}{a^3}\left(-\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}}\right) \qquad\mbox{(} n\not\in \{1, 2, 3\}\mbox{)}</math>

: <math>\int\frac{dx}{x(ax + b)} = -\frac{1}{b}\ln\left|\frac{ax+b}{x}\right|</math>

: <math>\int\frac{dx}{x^2(ax+b)} = -\frac{1}{bx} + \frac{a}{b^2}\ln\left|\frac{ax+b}{x}\right|</math>

: <math>\int\frac{dx}{x^2(ax+b)^2} = -a\left(\frac{1}{b^2(ax+b)} + \frac{1}{ab^2x} - \frac{2}{b^3}\ln\left|\frac{ax+b}{x}\right|\right)</math>

: <math>\int\frac{dx}{x^2+a^2} = \frac{1}{a}\arctan\frac{x}{a}\,\!</math>

: <math>\int\frac{dx}{x^2-a^2} = -\frac{1}{a}\,\mathrm{arctanh}\frac{x}{a} = \frac{1}{2a}\ln\frac{a-x}{a+x} \qquad\mbox{(}|x| < |a|\mbox{)}\,\!</math>

: <math>\int\frac{dx}{x^2-a^2} = -\frac{1}{a}\,\mathrm{arccoth}\frac{x}{a} = \frac{1}{2a}\ln\frac{x-a}{x+a} \qquad\mbox{(}|x| > |a|\mbox{)}\,\!</math>

: <math>\int\frac{dx}{ax^2+bx+c} = \frac{2}{\sqrt{4ac-b^2}}\arctan\frac{2ax+b}{\sqrt{4ac-b^2}} \qquad\mbox{(}4ac-b^2>0\mbox{)}</math>

: <math>\int\frac{dx}{ax^2+bx+c} = \frac{2}{\sqrt{b^2-4ac}}\,\mathrm{artanh}\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| \qquad\mbox{(}4ac-b^2<0\mbox{)}</math>

: <math>\int\frac{dx}{ax^2+bx+c} = -\frac{2}{2ax+b}\qquad\mbox{(}4ac-b^2=0\mbox{)}</math>

: <math>\int\frac{x}{ax^2+bx+c}dx = \frac{1}{2a}\ln\left|ax^2+bx+c\right|-\frac{b}{2a}\int\frac{dx}{ax^2+bx+c}</math>

: <math>\int\frac{mx+n}{ax^2+bx+c}dx = \frac{m}{2a}\ln\left|ax^2+bx+c\right|+\frac{2an-bm}{a\sqrt{4ac-b^2}}\arctan\frac{2ax+b}{\sqrt{4ac-b^2}} \qquad\mbox{(}4ac-b^2>0\mbox{)}</math>

: <math>\int\frac{mx+n}{ax^2+bx+c}dx = \frac{m}{2a}\ln\left|ax^2+bx+c\right|+\frac{2an-bm}{a\sqrt{b^2-4ac}}\,\mathrm{artanh}\frac{2ax+b}{\sqrt{b^2-4ac}} \qquad\mbox{(}4ac-b^2<0\mbox{)}</math>

: <math>\int\frac{mx+n}{ax^2+bx+c}dx = \frac{m}{2a}\ln\left|ax^2+bx+c\right|-\frac{2an-bm}{a(2ax+b)}\qquad\mbox{(}4ac-b^2=0\mbox{)}</math>

: <math>\int\frac{dx}{(ax^2+bx+c)^n} = \frac{2ax+b}{(n-1)(4ac-b^2)(ax^2+bx+c)^{n-1}}+\frac{(2n-3)2a}{(n-1)(4ac-b^2)}\int\frac{dx}{(ax^2+bx+c)^{n-1}}\,\!</math>

: <math>\int\frac{x}{(ax^2+bx+c)^n}dx = \frac{bx+2c}{(n-1)(4ac-b^2)(ax^2+bx+c)^{n-1}}-\frac{b(2n-3)}{(n-1)(4ac-b^2)}\int\frac{dx}{(ax^2+bx+c)^{n-1}}\,\!</math>

: <math>\int\frac{dx}{x(ax^2+bx+c)} = \frac{1}{2c}\ln\left|\frac{x^2}{ax^2+bx+c}\right|-\frac{b}{2c}\int\frac{dx}{ax^2+bx+c}</math>