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SolveByTangentSubstitution(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean)

Method (no overloads)

Summary

An integrand that is a function of tan(x) and of nothing else, integrated by
the substitution u = tan(x), under which dx is du/(1 + u^2).

Remarks

The substitution that turns every rational function of the tangent into a rational
function, and int sqrt(tan(x)) — the last of the five integrals
#233 lists — into
int sqrt(u)/(1 + u^2) du.
Why this is not a candidate in SolveBySubstitution(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean). That one
divides the integrand by du/dx and asks whether any x is left, which
works when the substitution survives the division. Here it does not:
sqrt(tan(x)) over the derivative of sqrt(tan(x)) is
2 tan(x) cos(x)^2, which is sin(2x) and is simplified to it — a correct
answer to a question that has stopped being about the tangent. Rewriting the integrand
asks a different question and does not lose the shape.
The test is the rewrite itself: replace every tan(x) and see whether an
x survives. tan(x) + x keeps one and is declined, which is right — it is
not a function of the tangent alone. The cotangent is covered, by writing it as
1/tan on the way in: it is the same function the other way up, and declining it
for having its own node meant tan(x)^2 was answered and cot(x)^2 was not.
No condition is owed by the substitution, but the answer inherits the
tangent's: u = tan(x) is undefined exactly where the integrand is, since the
integrand is a function of it, and 1 + u^2 is never zero for real u.

AngouriMath

Table of contents

AngouriMath

SolveByTangentSubstitution​(AngouriMath.​Entity,​AngouriMath.​Entity.​Variable,​System.​Boolean)

SolveByTangentSubstitution(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean)