Here is a table of circle trig substitutions and their equivalent hyperbolic trig substitutions. Integral Circle trig Hyperbolic trig substitution substitution dr r = asin(u) Va2 _ I = a tanh(u) - 12 r = a tan(u) I = a sinh(u) I = asec(u) I = a cosh(u) It is often simpler to use a substitution involving cos, sin or their hyperbolic equivalents cosh, sinh rather than using a tan, sec or tanh substitution. For example, to find the integral I = 1 1/ (2)2 We prefer to use the circle trig substitution = 2*sin(u) Since 2*cos(u) du & G, we can simplify this integral in terms of u: coS (u) I = du 1/1 - sin 2 (u) Now use the identity 1 - sin-(u) = cos(u)*2 GM to evaluate this integral first in terms of I = LI . G G +C. Then in terms of the original variable c: I = G 6 +C. Note: the Maple syntax for cos(x) is arccos (x) , and the syntax for tan-(x) is tan (x) *2