



When
you need to find the integral of the product



of
two or more trig functions, things get more complicated. 


Sometimes
you luck out and get something like this ... 











Then
you can just do a substitution ... 





u
= sinX du = cosXdX






That
gives you ... 











That
was an easy one. 





Here's
one that's not so easy ... 











What
would you do with this one? 


The
u and du trick won't work here, . 


because
the exponent on the cos is not right. 





The
trick we can use here, is to use the identity ... 





sin^{2}X
+ cos^{2}X = 1






With
this, we can change all of the powers of cosX except one into sinX, 


then
we can do the u and du substitution. 


Watch
... 











That
trick works great as long as one of the two 


(sinX
or cosX) is raised to an odd power. 





If
we had something like sin3Xcos4X, 


we
would have used the same identity to change 


all
but one of the sinX's to cosX and then let u = cosX and du = sinXdX. 


No
big deal. 





The
problem comes, when both sine and cosine 


are
raised to EVEN powers. 





Example: 











The
u and du trick won't work here. 


We
need a different trick. 


We
still use the same identity equation. 


But
here, we use it to change EVERYTHING 


into
either sine or cosine. 


You
can do whichever you like, it really doesn't matter. 


Let's
turn everything into sines ... 











Now
we need one of the reduction formulas from the last page. 


The
particular one we need is ... 











The
answer is going to be pretty long. 


Do
the two integrals as separate things ... 











and
... 











Putting
this all together and simplifying a bit, 


we
get ... 











You
might see a hint of a pattern here. 


You'll
see more of that pattern on the following pages! 





Remember
that any trig term can be turned into 


sines
and cosines ... 











and
will boil down to one of the problem types 


that
we just did. 





OR
...












If
the exponents on sinX and cosX are the same, 


we
are home free. 


These
puppies become ... 











Then
we can use the reduction formula for tanX or cotX. 





If
the exponents are NOT the same, we need another trick. 


Actually,
what we need is a new reduction formula or two. 





We
need ... 

















The
strategy with these puppies goes like this. 


If
the numerator exponent is even, 


go
through the reduction formula as many times as needed, 


to
make the numerator go away. 


The
you will be left with 1/(trig function). 


That
can be rewritten as either secant or cosecant. 


Then
use the reduction formula for that. 





BUT,
if the numerator exponent is odd, 


we
do something different. 


We
go through the reduction formula as many times as needed, 


to
make the numerator EXPONENT be one. 


Then
we use u and du substitution. 


The
numerator will be du, 


and
we will wind up with ln(something) for the last term. 





Example: 











Simplify
... 











In
the integral part, substitute ... 





u
= sinX du = cosXdX












Simplify
... 











u
= sinX, so ... 











You
can find a common denominator, 


and
simplify from here if you want. 





OK,
that was one with an odd power of sine or cosine 


in
the numerator. 


Now
let's do one with an even power of sine or cosine 


in
the numerator ... 





Example: 











Simplify
a bit ... 











Use
the secant reduction formula ... 











copyright 2005 Bruce Kirkpatrick 
