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We're Not In Triangles Any More, Toto
Trig Integrals

 

 Finding the integral of the sine and cosine functions is easy.

 We know the derivative of the sine is the cosine,
 so the integral of the cosine is the sine.
 

 
 We also know that the derivative of the cosine
 is the negative sine (-sin).
 That means that the integral of the sine is negative cosine.
 

 
 The integrals of the other four trig functions
 are not quite as obvious.
 To find the integral of the tangent, 
 remember that the tangent is equal to
 the sine divided by the cosine.
 

 
 
 The sine divided by cosine thing is easier to deal with
 than you might think.
 We let ...
 

 u = cosX    du = - sinXdx   (so - du = sinXdx)

 
 That gives us ...
 

 
 A basic property of logarithms is that if we have the log (or ln)
 of something raised to a power,
 we can turn the power into a coefficient of the log.
 We can also do the reverse.
 So ...
 

 
 Using this idea on our -ln|cosX| term we get ...
 

 
 so the integral of tanX is ...
 

 
 Take your choice, they're both the same value!
 Finding the integral of the cotangent is similar ...
 

 
 See that this one is simpler than the tangent
 because we don't have any "-" signs to deal with anywhere.
 The math type who figured out what the integral
 of the secant was,
 must have worked on it for a long time.
 
 The idea was to multiply the integral of secXdx
 by something to make it into ...
 

 
 or something like it.
 
 What was hit upon was ...
 

 
 since the derivative of secX is secXtanX 
 and the derivative of tanX is sec 2X,
 we have u in the denominator and du in the numerator.
 This is the "ln" integral form,
 so ...
 

 
 The integral of the cosecant function is found in a very similar way.
 

 
 We can replace X by more complex things
 and still find the integral.
 All we need is to have the right dx,
 or something that we can make into the right dx.
 
 For example, if we want to find the integral of csc6X ...
 

 
 We need a 6 (the proper coefficient for dX) to do the integral,
 so we add a 6 1/6 to the integral to get the right dX.
 

 
 Using our formula we have ...
 

 
 We can use this strategy on most integrals.
 To show this, we usually use "u" instead of X
 when we write a general form of an integral.
 The general form of the integral equations
 of our six trig functions are ...
 

 

   copyright 2005 Bruce Kirkpatrick

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