universal substitution (idea)

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.