The universal
substitution is a handy tool for evaluating
integrals
that consist of a
rational function of
trigonometric functions.
This is the substitution:
u = tan(x/2), if originally integrating with respect to x.
You will want to substitute in for dx:
u = tan(x/2)
x/2 = arctan(u)
x = 2*arctan(u)
dx = 2*du/(1+u^2)
The substitutions for trigonometric functions:
sin(x) = 2*sin(x/2)*cos(x/2)
= 2*sin(x/2)*cos(x/2)^2/cos(x/2)
= 2*tan(x/2)*cos(x/2)^2
= 2*tan(x/2)/sec(x/2)^2
= 2*tan(x/2)/(1+tan(x/2)^2)
sin(x) = 2*u/(1+u^2)
cos(x)^2 = 1 - sin(x)^2
= 1 - (2*u/(1+u^2))^2
= 1 - 4*u^2/(1+u^2)^2
= ((1 + u^2)^2 - 4*u^2)/(1+u^2)^2
= (1 + 2*u^2 + u^4 - 4*u^2)/(1+u^2)^2
= (1 - 2*u^2 + u^4)/(1+u^2)^2
= (1-u^2)^2/(1+u^2)^2
cos(x) = (1-u^2)/(1+u^2)
Others are easy now. For example,
tan(x) = sin(x)/cos(x)
= 2*u/(1-u^2)
Once you have substituted in for the trigonometric functions,
you should get a rational
function in terms of
u. This
can usually be
simplified further. However, you may next need to
go on to
partial fraction decomposition.
And remember kids, make sure you substitute back in for u when you are done.