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There is one function where our integral formula does not work. |
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| If we have the equation: | ||||||
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| Then applying the rule would give us: | ||||||
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| WOW! Try to make sense out of that! | ||||||
| On second thought, don't. | ||||||
| The real answer is: | ||||||
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| OK, What's an "ln" ??? | ||||||
| "ln" Lower Case "LN" is actually a special type of logarithm, | ||||||
| with a base of about 2.71828. | ||||||
| The actual base number is a special irrational number. | ||||||
| It is so special, that it gets it's own letter for a name: "e" | ||||||
| This "e" thing has all kinds of applications, and we'll flog them to pieces in time. | ||||||
| For now, just remember that: | ||||||
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| Why do we need the absolute value sign??? | ||||||
| The way logarithms work, | ||||||
| "lnX" means the exponent that 2.71828 needs to be raised to, to get X. | ||||||
| No matter what power you raise 2.71828 to, you always get a positive number, | ||||||
| so we use the absolute value sign to make sure we know | ||||||
| to have a positive value for X. | ||||||
| OK, Fine. But what if we need the integral of some complex function like: | ||||||
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F(x) = (3X - 5)-1 |
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| What then??? | ||||||
| Well then we're back to the old outside/inside thing. | ||||||
| We need to have the derivative of the inside part sitting there at the start. | ||||||
| We have | ||||||
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| And we need: | ||||||
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| dx is just 1 so that's no problem, | ||||||
| but we need our little change the du constant trick again... | ||||||
| So: | ||||||
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| The general form of this "ln" thing is: | ||||||
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| By the way, even though we're not really looking for derivatives right now | ||||||
| we've stumbled on the fact that the derivative of lnX is 1/X. | ||||||
| Hey, take what you can get! | ||||||
| Another Example: | ||||||
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copyright 2005 Bruce Kirkpatrick |
