Difference between revisions of "Table of derivatives general rules" - Rhea

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-[[Category:Formulas]]
-[[Category:integral]]
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-! style="background-color: rgb(228, 188, 126); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | Table of Infinite Integrals
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-! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | 1 General Rules
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-|<math> \int a d x =  a x </math>
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-|<math> \int a f ( x ) d x = a \int f ( x ) d x </math>
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-|<math> \int ( u \pm v \pm w \pm \cdot \cdot \cdot ) d x = \int u d x \pm \int v d x \pm \int w d x \pm \cdot \cdot \cdot</math>
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-|<math> \int u d v = u v - \int v d u </math>
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-|<math> \int f ( a x ) d x = \frac{1}{a} \int f ( u ) d u </math>
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-|<math> \int F \{ f ( x ) \} d x = \int F ( u ) \frac{dx}{du} d u = \int \frac{F ( u )}{f^' ( x )} d u \qquad  u = f ( x ) </math>
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-|<math> \int u^n d u = \frac{u^{n+1}}{n+1} \qquad n \neq -1 </math>
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-|<math> \int \frac{d u}{u} =  \ln u \ ( if \ u > 0 ) \ \text{or} \ln {-u} \ ( \text{if} \ u < 0 ) = \ln \left | u \right | </math>
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-|<math> \int e^u d u = e^u </math>
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-|<math> \int a^u d u = \int e^{u \ln a} d u = \frac{e^{u \ln a}}{\ln a} = \frac{a^u}{\ln a} \qquad a > 0 \ \text{and} \ a \neq 1</math>
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-|<math> \int \sin u\ d u = - \cos u </math>
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-|<math> \int \cos u\ d u =  \sin u </math>
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-|<math> \int \tan u\ d u =  - \ln {\cos u} </math>
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-|<math> \int \cot u\ d u =   \ln {\sin u} </math>
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-|<math> \int \frac{d u}{\cos u} =   \ln { \left ( \frac{1}{\cos u} + \tan u \right )}  = \ln{\tan   {\left ( \frac{u}{2}+\frac{\pi}{4}\right )}} </math>
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-|<math> \int \frac{d u}{\sin u} =   \ln { \left ( \frac{1}{\sin u} - \cot u \right )}  = \ln{\tan   { \frac{u}{2}}} </math>
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-|<math> \int \frac{d u}{\cos ^2 u} = \tan u </math>
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-|<math> \int \frac{d u}{\sin ^2 u} = - \cot u </math>
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-|<math> \int \tan ^2 u \ d u =  \tan u - u</math>
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-|<math> \int \cot ^2 u \ d u =  - \cot u - u</math>
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-|<math> \int \sin ^2 u  \ d u=  \frac{u}{2} - \frac{\sin {2 u}}{4} = \frac{1}{2}\left( u - \sin u \cos u \right )</math>
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-|<math> \int \frac {1}{\cos  u} \tan u \ d u =  \frac{1}{\cos u}</math>
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-|<math> \int \frac {1}{\sin  u} \cot u \ d u = - \frac{1}{\sin u}</math>
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-|<math> \int \sinh u \ d u =  \coth u</math>
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-|<math> \int \cosh u \ d u =  \sinh u</math>
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-|<math> \int \tanh u \ d u =  \ln \cosh u</math>
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-|<math> \int \coth u \ d u = \ln \sinh u</math>
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-|<math> \int \frac {1}{\operatorname{ch}\ u}  \ d u = \arcsin{\left ( \operatorname{th}\,u \right )} \qquad or \  2 arc \ th \  e^u</math>
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-|<math> \int \frac {1}{\operatorname{sh}\ u}  \ d u = \ln \operatorname{th}\,\frac{2}{2} \qquad or \ - \operatorname{Arg coth} \ e^u</math>
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-|<math> \int \frac {1}{\operatorname{ch^2}\ u}  \ d u =  \operatorname{th}\,u  </math>
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-|<math> \int \frac {1}{\operatorname{sh^2}\ u}  \ d u = - \operatorname{coth}\,u  </math>
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-|<math> \int \operatorname{th^2}\ u \ d u = u - \operatorname{th}\,u  </math>
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-|<math> \int \operatorname{coth^2}\ u \ d u = u - \operatorname{coth}\,u  </math>
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-|<math> \int \operatorname{sh^2}\ u \ d u = \frac {\operatorname{sh}\,{2 u}}{4} - \frac{u}{2}=\frac{1}{2}\left ( \operatorname{sh}\,u \ \operatorname{ch}\,u  - u \right )</math>
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-|<math> \int \operatorname{ch^2}\ u \ d u = \frac {\operatorname{sh}\,{2 u}}{4} + \frac{u}{2}=\frac{1}{2}\left ( \operatorname{sh}\,u \ \operatorname{ch}\,u  + u \right )</math>
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-|<math> \int \frac{\operatorname th \ u}{\operatorname ch \ u} \ d u = - \frac {1}{\operatorname ch \, u }</math>
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-|<math> \int \frac{\operatorname coth \ u}{\operatorname sh \ u} \ d u = - \frac {1}{\operatorname sh \, u }</math>
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-|<math> \int \frac{d u}{u^2 + a^2} =  \frac {1}{a}\arctan \frac{u}{a}</math>
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-|<math> \int \frac{d u}{u^2 - a^2} =  \frac {1}{2 a}\ln \left ( \frac{u-a}{u+a} \right ) = -\frac{1}{a} \operatorname{argcoth} \ \frac{u}{a} \qquad u^2 > a^2 </math>
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-|<math> \int \frac{d u}{a^2 - u^2} =  \frac {1}{2 a}\ln \left ( \frac{a+u}{a-u} \right ) = \frac{1}{a} \operatorname{argth}\ \frac{u}{a} \qquad u^2 < a^2 </math>
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-|<math> \int \frac{d u}{\sqrt{a^2 - u^2}} = \arcsin  \frac{u}{a}  </math>
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-|<math> \int \frac{d u}{\sqrt{u^2 + a^2}} = \ln { \left ( u + \sqrt {u^2+a^2} \right ) } \qquad or \ \operatorname{argth} \ \frac{u}{a}  </math>
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-|<math> \int \frac{d u}{\sqrt{u^2 - a^2}} = \ln { \left ( u + \sqrt {u^2-a^2} \right ) } </math>
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-|<math> \int \frac{d u}{u \sqrt{u^2 - a^2}} = \frac {1}{a} \arccos \left | \frac{a}{u} \right |  </math>
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-|<math> \int \frac{d u}{u \sqrt{u^2 + a^2}} = - \frac {1}{a} \ln \left ( \frac{a + \sqrt{u^2 + a^2}}{u} \right )   </math>
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-|<math> \int \frac{d u}{u \sqrt{a^2 - u^2}} = - \frac {1}{a} \ln \left ( \frac{a + \sqrt{a^2 - u^2}}{u} \right )   </math>
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-|<math> \int f^{(n)} \ g d x  =f^{(n-1)} \ g - f^{(n-2)} \ g' + f^{(n-3)} \ g'' - \cdot \cdot \cdot \ (-1)^n \int fg^{(n)} d x  </math>
-|}
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-[[Table_of_indefinite_integrals|Back to Table of Indefinite Integrals]]
-[[Collective_Table_of_Formulas|Back to Collective Table of Formulas]]

Difference between revisions of "Table of derivatives general rules" - Rhea

Latest revision as of 11:33, 17 November 2013

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