



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 sine^{2} 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 
