查看“︁积分列表”︁的源代码


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{{Wikipedia|积分}}

以下列出一些常用的积分。
<!-- 帮忙改好这句吧! -->

=== 符号意义 ===

为简便见，以下<math>n</math>和<math>c</math>代表一个[[常数]]、<math>x</math>是[[变数]]、<math>f</math>和 <math>g</math>是<math>x</math>的[[函数]]。不定积分的[[常数项]]將不会列出。

==不定积分==

=== [[多项式]]、[[指数]]及[[对数]] ===

*<math>\int {c\;dx}  = cx</math>
*<math>\int {x^c dx}  = {{x^{c + 1} } \over {c + 1}}</math> (若 ''c'' ≠- 1)
*<math>\int {{1 \over x}dx}  = \ln \left| x \right|</math>
&nbsp;
*<math>\int {e^x dx}  = e^x</math>
*<math>\int {c^x dx}  = {{c^x } \over {\ln c}}</math>
*<math>\int {\ln x\;dx}  = x\left( {\ln x - 1} \right)</math>
*<math>\int {\log _c x\;dx}  = {{x\left( {\ln x - 1} \right)} \over {\ln c}}</math>
*<math>\int {x^n e^{-x} dx}  = -[x^n+nx^{n-1}+n(n-1)x^{n-2}+n(n-1)(n-2)x^{n-3}+ \cdots + n!x^0]e^{-x}, n\in N</math>

=== [[三角函数]]及反三角函数 ===

*<math>\int {\sin x\;dx}  =  - \cos x</math>
*<math>\int {\cos x\;dx}  = \sin x</math>
*<math>\int {\tan x\;dx}  = \ln \left| {\sec x} \right|</math>
*<math>\int {\cot x\;dx}  = \ln \left| {\sin x} \right|</math>
*<math>\int {\sec x\;dx}  = \ln \left| {\sec x + \tan x} \right|</math>
*<math>\int {\csc x\;dx}  = \ln \left| {\tan {x \over 2}} \right|</math>
&nbsp;
*<math>\int {\sin ^2 x\;dx}  = {{2x - \sin 2x} \over 4}</math>
*<math>\int {\cos ^2 x\;dx}  = {{2x + \cos 2x} \over 4}</math>
*<math>\int {\tan ^2 x\;dx}  = \tan x - x</math>
*<math>\int {\cot ^2 x\;dx}  =  - \cot x - x</math>
*<math>\int {\sec ^2 x\;dx}  = \tan x</math>
*<math>\int {\csc ^2 x\;dx}  =  - \cot x</math>
&nbsp;
*<math>\int {\sin ^3 x\;dx}  = {{\cos ^3 x - 3\cos x} \over 3}</math>
*<math>\int {\cos ^3 x\;dx}  = {{3\sin x - \sin ^3 x} \over 3}</math>
*<math>\int {\tan ^3 x\;dx}  = {{\sec ^2 x - \ln \sec ^2 x} \over 2}</math>
*<math>\int {\cot ^3 x\;dx}  = {{\ln \csc ^2 x - \csc ^2 x} \over 2}</math>
&nbsp;
*<math>\int {\sin ^{ - 1} x\;dx}  = x\sin ^{ - 1} x + \sqrt {1 - x^2 } </math>
*<math>\int {\cos ^{ - 1} x\;dx}  = x\cos ^{ - 1} x - \sqrt {1 - x^2 } </math>
*<math>\int {\tan ^{ - 1} x\;dx}  = x\tan ^{ - 1} x - \ln \sqrt {1 + x^2 } </math>
*<math>\int {\cot ^{ - 1} x\;dx}  = x\cot ^{ - 1} x + \ln \sqrt {1 + x^2 } </math>

=== [[双曲函数]]及反双曲函数 ===

*<math>\int {\sinh x\;dx}  =  \cosh x</math>
*<math>\int {\cosh x\;dx}  = \sinh x</math>
*<math>\int {\tanh x\;dx}  = \ln {\mathop{\rm sech}} x</math>
*<math>\int {\coth x\;dx}  = \ln \left| {\sinh x} \right|</math>
*<math>\int {{\mathop{\rm sech}} x\;dx}  = \sin ^{ - 1} \tanh x = 2\tan ^{ - 1} e^x 
</math>
*<math>\int {{\mathop{\rm csch}} x\;dx}  = \ln \left| {\tanh {x \over 2}} \right| =  - 2\coth ^{ - 1} e^{\left| x \right|} </math>
&nbsp;
*<math>\int {\sinh ^{ - 1} x\;dx}  = x\sinh ^{ - 1} x - \sqrt {x^2  + 1} </math>
*<math>\int {\cosh ^{ - 1} x}  = x\cosh ^{ - 1} x \mp \sqrt {x^2  - 1} </math>

=== 其它形式 ===

*<math>\int {{{dx} \over {n^2 x^2  + c^2 }}}  = {1 \over {nc}}{\tan ^{ - 1} {{nx} \over c}}</math>
*<math>\int {{{dx} \over {n^2 x^2  - c^2 }}}  =  - {1 \over {nc}}{\tanh ^{ - 1} {{nx} \over c}}</math>
&nbsp;
*<math>\int {{{dx} \over {\sqrt {x^2  + c^2 } }}}  = \sinh ^{ - 1} {x \over c}</math>
*<math>\int {{{dx} \over {\sqrt {x^2  - c^2 } }}}  = \ln \left( {x + \sqrt {x^2  - c^2 } } \right)</math>
*<math>\int {{{dx} \over {\sqrt {c^2  - x^2 } }}}  = \sin ^{ - 1} {x \over c}</math>
&nbsp;
*<math>\int {{{dx} \over {x\sqrt {x^2  + c^2 } }}}  =  - {1 \over c}\ln \left( {{{c + \sqrt {c^2  + x^2 } } \over x}} \right)</math>
*<math>\int {{{dx} \over {x\sqrt {x^2  - c^2 } }}}  =  - {1 \over c}\sec ^{ - 1} {x \over c}</math>
*<math>\int {{{dx} \over {x\sqrt {c^2  - x^2 } }}}  =  - {1 \over c}\ln \left( {{{c + \sqrt {c^2  - x^2 } } \over x}} \right)</math>

=== [[分部积分法]] ===

*<math>\int {fg'\;dx}  = fg - \int {f'g\;dx} </math>

== 定积分 ==
*<math>\int_0^{+\infty} \frac{\sin x}{x} dx = \frac{\pi}{2}</math>
*<math>\int_0^{+\infty} \frac{\sin^2 x}{x^2} dx = \frac{\pi}{2}</math>
*<math>\int_0^{+\infty} x^z e^{-\lambda x} dx = \lambda^{-(z+1)}\Gamma(z+1) </math> (其中<math>\lambda>0</math>)
*<math>\int_{-1}^{1}e^{izx} dx =\frac{2\sin z}{z}</math>
*<math>\int_0^{\pi/2}\cos^ax\ \sin^bx\ dx = \frac{1}{2}B(\frac{a+1}{2},\frac{b+1}{2})</math> （其中<math>a, b>-1</math>, <math>B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}</math>）

[[Category:数学]]