Appendix B
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Short appendix, dated 1.11.15 (apparently January 2015), from Phil's study of the integral of 1/sqrt(R). It lists Euler substitutions A, B and C, each with two sign variants, giving the transformed integral and the formula for t(x). Substitution C uses the factored form c(x-α)(x-β). Equation numbers (4.7)-(6.8) refer to the main text. Most symbols were lost in extraction, so the formulas are not readable.
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This is the Title PhL 1.11.15
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Appendix B. Summary of the three Euler substitutions
Substitution A t ± x =
!Syntax Error, Idx R(x,) = !Syntax Error, Idt 2 * R( , )
where t(x) = - x . (4.7)
!Syntax Error, Idx R(x,) = !Syntax Error, Idt 2 * R( , )
where t(x) = + x . (4.8)
Substitution B xt ± =
!Syntax Error, Idx R(x,) = !Syntax Error, Idt 2 * R( , )
where t(x) = ( - ) / x . (5.7)
!Syntax Error, Idx R(x,) = !Syntax Error, Idt 2 * R( , )
where t(x) = ( + ) / x . (5.8)
Substitution C : t = ()±1 where a+bx+cx2 = c(x-α)(x-β)
!Syntax Error, Idx R(x,) = !Syntax Error, Idt 2 * R(, )
where t(x) = / (x-α) = /(x-α) . (6.7)
!Syntax Error, Idx R(x,) = !Syntax Error, Idt 2 * R(, ) (6.8)
where t(x) = / (x-β) = /(x-β) .