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Nasty Trig Stuff
Integrating Composite Trig Functions

 

 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 ...
 

 sin2X + cos2X = 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

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