Reduction formulae
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Reduction formulae
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We find [Equation goes here - download the original to see it.] by means of the trigonometric identity [Equation goes here - download the original to see it.] For higher powers of cosn x, i.e. for n > 4, the use of trigonometric identities becomes tedious. Fortunately, a further technique exists - the use of a reduction formula. When a reduction formula is used we rewrite the integral in, say, cosn x in terms of some power less than n. We then repeat the process. The method is best illustrated by example. [Equation goes here - download the original to see it.] We integrate cos4 x by parts. The parts formula is:- [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Then [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] We continue this investigation by examining [Equation goes here - download the original to see it.] As before [Equation goes here - download the original to see it.] and we will begin by integrating by parts [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Substituting [Equation goes here - download the original to see it.] and collecting [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Let us summarise what we have discovered so far regarding [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Let us now examine the definite integrals of the form [Equation goes here - download the original to see it. [Equation goes here - download the original to see it.] From this we can form the conjecture: [Equation goes here - download the original to see it.] This is an example of a reduction formula because it evaluates an integral of a certain type in an expression to the power n in terms of an integral of the same type but in a reduced power. Repeated applications of a reduction formula will eventually reduce an integral to found to a known integral. Thus, here [Equation goes here - download the original to see it.] We will now prove the validity of the conjecture. [Equation goes here - download the original to see it.] We will now integrate this by parts. [Equation goes here - download the original to see it.] Questions on reduction formula ask you to prove such relationships as [Equation goes here - download the original to see it.] - a process that will generally involve integration by parts - and ask you to apply the result to find a definite integral, as in [Equation goes here - download the original to see it.]
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Contents of
Reduction formulae
1 Reduction formulae
2 Simultaneous differential equations - Systems of differential equations
3 Matrix notation- Simultaneous differential equations - Systems of differential equat
4 First order systems - Simultaneous differential equations - Systems of differential
5 Example - Simultaneous differential equations - Systems of differential equations
6 Solution to first-order constant coefficient homogeneous systems
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