I happened upon this in MIT OpenCourseWare:
This transform allows converting an integral of the form (tan(x))^n to a polynomial problem.
f( tan x ) = ( tan x )^4
y = tan x
f( y ) = ( y )^4
Substituting into the transform:
Integral [tan^4 (x) dx] = Integral [ y^4 / (1 + y^2) dy ]
Using long division:
y^4 / (1 + y^2) = y^2 - 1 + 1 / (1 + y^2)
Substituting into the right-hand-side of the transformed problem above and performing the integration:
Integral [ y^2 - 1 + 1 / (1 + y^2) dy ] = 1/3 * y^3 - y + Arctan(y)
Then returning to the x-world:
Integral [tan^4 (x) dx] = 1/3 * tan^3(x) - tan(x) + x + C
He said in the video: "there's a whole family of things like that," but I don't know where.
My Google searches didn't point me to a list where this transform is included.
Are other members of this family available online somewhere?
Maybe this family could be printed on T-shirts?