



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 lncosX 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
^{2}X, 


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 
