AngouriMath
TheFirstPowerCancels(AngouriMath.Entity,AngouriMath.Entity,AngouriMath.Entity,AngouriMath.Entity,AngouriMath.Entity,System.Int32,AngouriMath.Entity.Variable)
Method (no overloads)
Summary
Whether, for a quadratic whose coefficients are real numbers with a nonzero leading
one and a nonzero discriminant, the first power's antiderivative -- the logarithm or
the arctangent every reduction above bottoms out on -- has coefficient zero once the
pieces of ∫ N(x)/Q^n are added up.
one and a nonzero discriminant, the first power's antiderivative -- the logarithm or
the arctangent every reduction above bottoms out on -- has coefficient zero once the
pieces of ∫ N(x)/Q^n are added up.
Remarks
The reduction answers each constant over a power of the quadratic on its own, down
to that first power, so a numerator of degree four over a fourth power arrives as
three reductions each carrying the logarithm, with coefficients that sum to zero
where the antiderivative is rational:8(x^4 + x^2)/(x^2 - 1)^4 is
-8x^3/(3(x^2 - 1)^3) , and it was written with ln((x - 1)/(x + 1)) three
times over. Correct, and read by nothing: integration by parts against it -- it is
cosh(x)/sinh(x)^4 under u = e^x , and Timofeev's
arccot(cosh x) cosh x/sinh^4 x is a step of parts against that -- spent four
seconds simplifying the logarithms out of the remainder. The coefficient is a number
and is added up here before anything is written; the writing itself is unchanged.
to that first power, so a numerator of degree four over a fourth power arrives as
three reductions each carrying the logarithm, with coefficients that sum to zero
where the antiderivative is rational:
times over. Correct, and read by nothing: integration by parts against it -- it is
seconds simplifying the logarithms out of the remainder. The coefficient is a number
and is added up here before anything is written; the writing itself is unchanged.
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