Antiderivative: Difference between revisions
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* add 1 to the power <math>3</math>, so that it is now <math>(2x+4)^4</math> |
* add 1 to the power <math>3</math>, so that it is now <math>(2x+4)^4</math> |
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* divide all this by the new power to get <math>\frac{(2x+4)^4}{4}</math> |
* divide all this by the new power to get <math>\frac{(2x+4)^4}{4}</math> |
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* |
* divide all this by the [[derivative]] of the bracket <math>\left (\frac{d(2x+4)}{dx} = 2 \right )</math> to get <math>\frac{(2x+4)^4}{4 \times 2} = \frac{1}{8}(2x+4)^4</math> |
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* and add a constant <math>c</math> to give <math>\frac{1}{8}(2x+4)^4 + c</math> |
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=== Examples === |
=== Examples === |
Revision as of 16:50, 14 June 2008
The English used in this Difficult subject to explain simply. may not be easy for everybody to understand. |
Antidifferentiation (or indefinite integration) is a part of mathematics. It is the opposite of differentiation. It is integrating with no limits. The answer is an equation.
It is written as
- with the integral sign that has no limits
- the equation you are integrating
- and the which means "with respect to ", which does not mean anything with simple integration.
Simple integration
To integrate
- add 1 to the power , so is now
- divide all this by the new power, so it is now
- and a constant should be added, so it is now
This can be shown as:
When there is many terms, integrate each part on its own:
(This only works if the parts are being added or taken away.)
Examples
Changing fractions and roots into powers makes it easier:
Integrating a bracket ("chain rule")
If you want to integrate a bracket like , we need to do it a different way. It is called the chain rule. It is like simple integration. It only works if the in the bracket has a power of 1 (it is linear) like or (not or ).
To do
- add 1 to the power , so that it is now
- divide all this by the new power to get
- divide all this by the derivative of the bracket to get
- and add a constant to give
Examples