From 49f5fd91374cc2e69ffa85936ea91fee168e6b66 Mon Sep 17 00:00:00 2001 From: kleinbottle Date: Sun, 10 Dec 2023 11:31:43 +0530 Subject: [PATCH] Changed 'integration' to 'differentiation' --- quarto/integrals/partial_fractions.qmd | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/quarto/integrals/partial_fractions.qmd b/quarto/integrals/partial_fractions.qmd index cd9b4570..e72ab07b 100644 --- a/quarto/integrals/partial_fractions.qmd +++ b/quarto/integrals/partial_fractions.qmd @@ -14,7 +14,7 @@ using SymPy Integration is facilitated when an antiderivative for $f$ can be found, as then definite integrals can be evaluated through the fundamental theorem of calculus. -However, despite integration being an algorithmic procedure, integration is not. There are "tricks" to try, such as substitution and integration by parts. These work in some cases. However, there are classes of functions for which algorithms exist. For example, the `SymPy` `integrate` function mostly implements an algorithm that decides if an elementary function has an antiderivative. The [elementary](http://en.wikipedia.org/wiki/Elementary_function) functions include exponentials, their inverses (logarithms), trigonometric functions, their inverses, and powers, including $n$th roots. Not every elementary function will have an antiderivative comprised of (finite) combinations of elementary functions. The typical example is $e^{x^2}$, which has no simple antiderivative, despite its ubiquitousness. +However, despite differentiation being an algorithmic procedure, integration is not. There are "tricks" to try, such as substitution and integration by parts. These work in some cases. However, there are classes of functions for which algorithms exist. For example, the `SymPy` `integrate` function mostly implements an algorithm that decides if an elementary function has an antiderivative. The [elementary](http://en.wikipedia.org/wiki/Elementary_function) functions include exponentials, their inverses (logarithms), trigonometric functions, their inverses, and powers, including $n$th roots. Not every elementary function will have an antiderivative comprised of (finite) combinations of elementary functions. The typical example is $e^{x^2}$, which has no simple antiderivative, despite its ubiquitousness. There are classes of functions where an (elementary) antiderivative can always be found. Polynomials provide a case. More surprisingly, so do their ratios, *rational functions*.