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bennybob12
Dec 29, 2019
  ·  Edited: Jan 01, 2020

Transforming Integral of [(tan(x))^4 dx] to a polynomial problem

in Video Ideas

I happened upon this in MIT OpenCourseWare:



https://www.youtube.com/watch?v=PNKj529yY5c&feature=youtu.be&t=963


This transform allows converting an integral of the form (tan(x))^n to a polynomial problem.


For example:


Let:

f( tan x ) = ( tan x )^4

and

y = tan x


Then:

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?

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