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<math>\int \frac{1}{(7x+12)^9}\ dx = \int (7x+12)^{-9}\ dx = \frac{(7x+12)^{-8}}{-8 \times 7} + c = -\frac{1}{54}(7x+12)^{-8} + c = -\frac{1}{54(7x+12)^8} + c \left ( \because \frac{d(7x+12)}{dx} = 7 \right )</math> |
<math>\int \frac{1}{(7x+12)^9}\ dx = \int (7x+12)^{-9}\ dx = \frac{(7x+12)^{-8}}{-8 \times 7} + c = -\frac{1}{54}(7x+12)^{-8} + c = -\frac{1}{54(7x+12)^8} + c \left ( \because \frac{d(7x+12)}{dx} = 7 \right )</math> |
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== Other pages == |
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* [[Mathematics]] |
* [[Mathematics]] |
Revision as of 04:13, 19 July 2009
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 are 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