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Atkins For Trig Integrals
Trig Integral Reduction Formulas

 

 When we have a sine or cosine raised to some power,

 it makes finding an integral more involved.
 
 One strategy, is to use identities to change the term
 into something that doesn't have an exponent.
 (other than 1 that is ...)
 
 If we have something like ...
 

 
 We can look in our list of trig identities
 and come up with these double angle formulas ...
 

 
 Let's work out the sine2 integral and see what happens ... 
 

 
 That wasn't TOO bad.
 But what if the exponents get much bigger?
 This could get very messy.
 
 It will get messy, but there is help out there from something called:
 

REDUCTION FORMULAS

 
 The great thing about these reduction formulas,
 is that they will work with any power of trig function
 that is greater than one.
 Here they are ...
 

 

 
 When we use one of these formulas on a trig integral
 that has an exponent greater than two,
 we will still be left with an integral on the end to do.
 
 If the term on the end is to the first power,
 we can use one of our easier ways to solve the integral.
 
 If the term on the end has an exponent greater than one,
 We just use the formula over again 
 until we are left with an integral of a term to the first power,
 or no integral at all.
 
 Example:
 
 Suppose we have ...
 

 
 The first time we put this through the reduction formula,
 we get ...
 

 

 
 We still have an integral on the end.
 The term has an exponent greater than one.
 That means we take another pass through the formula.
 

 
 We still have an integral on the end.
 The term still has an exponent greater than one.
 That means we need to make another pass with the formula ...
 

 
 We still have an integral on the end,
 but the term DOES NOT have an exponent greater than one.
 That means we can use one of our regular trig integral formulas
 
 
 And we're finally done.
 
 These things aren't particularly hard,
 you just have to "turn the crank" on them for a while.
 

   copyright 2005 Bruce Kirkpatrick

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