AngouriMath

Navigation

Home Quick start Wiki Almanac Research What's new Try Use Angouri.org GitHub Report a bug ~~Donate~~ Contacts

ByPartsAgainstThePowersOfX(System.Collections.Generic.IReadOnlyList{AngouriMath.Entity},AngouriMath.Entity.Variable)

Method (no overloads)

Summary

A sum whose terms are not elementary apart, integrated by parts against the power
of x in each term that has one, with what parts leaves and the
other terms asked together as one integrand: x^k f + g is
x^k F + ∫(g - k x^(k - 1) F) for F the antiderivative of f.

Remarks

sqrt(csc(x)) (x cos(x) - 4 sec(x) tan(x)), Rubi's 4.7.7, is
(2 x sqrt(sin(x)) - 4 sin(x)^(3/2)/cos(x))'. Split into its two terms, neither is
elementary: parts on the first leaves -2 sqrt(sin(x)), and that and the second
term are elementary only together, as sqrt(sin(x)) (-2 - 4/cos(x)^2). So the
remainder is gathered into one quotient with the factors its terms share taken out,
which is the spelling the rules read; and where it is zero, as for
x cos(x) e^sin(x) + e^sin(x), the parts are the answer.
Only where some term with a power of x beside something else in
x has an elementary F, and only at the question asked: the
remainder is a new question, and the powers of x of the terms
integrated are lower in it, so asking it again descends.

AngouriMath

Table of contents

AngouriMath

ByPartsAgainstThePowersOfX​(System.​Collections.​Generic.​IReadOnlyList{AngouriMath.​Entity},​AngouriMath.​Entity.​Variable)

ByPartsAgainstThePowersOfX(System.Collections.Generic.IReadOnlyList{AngouriMath.Entity},AngouriMath.Entity.Variable)