2117: Differentiation and Integration - Revision history

SomebodyElse: /* Explanation */ just linking characters' names

2026-05-13T12:33:51Z

‎Explanation: just linking characters' names

199.79.156.33: /* Explanation */

2025-11-02T18:43:13Z

‎Explanation

Lettherebedarklight: /* Explanation */ +

2025-04-20T06:14:23Z

‎Explanation: +

162.158.79.192: removed Category:Emoji

2024-12-31T04:23:13Z

removed Category:Emoji

172.68.225.160 at 02:16, 14 December 2024

2024-12-14T02:16:46Z

162.158.91.91: /* Integration */ spaces around dots

2024-08-07T01:52:00Z

‎Integration: spaces around dots

162.158.90.177: /* Differentiation */ might as well put spaces around the dots

2024-08-07T01:50:37Z

‎Differentiation: might as well put spaces around the dots

162.158.90.176: /* Integration */ dots instead of implicit multiplication

2024-08-07T01:49:58Z

‎Integration: dots instead of implicit multiplication

172.68.23.73: /* Integration */ comma

2024-08-07T01:47:54Z

‎Integration: comma

162.158.91.91: /* Differentiation */ consistency

2024-08-07T01:46:53Z

‎Differentiation: consistency

@@ Line 12: / Line 12: @@
 {{w|Derivative|Differentiation}} and {{w|Antiderivative|Integration}} are two major components of {{w|calculus}}. As many Calculus 2 students are painfully aware, integration is much more complicated than the differentiation it undoes.
-However, Randall dramatically overstates this point here.  After the first step of integration, Randall assumes that any integration can not be solved so simply, and then dives into a step named "????", suggesting that it is unknowable how to proceed.  The rest of the flowchart is (we can assume deliberately) even harder to follow, and does not reach a conclusion.  This is in contrast to the simple, straightforward flowchart for differentiation. The fact that the arrows in the bottom of the integration part lead off the bottom of the page indicates that "Phone calls to mathematicians", "Oh no" and "Burn the evidence" are not final steps in the difficult journey. The flowchart could be extended by Randall to God-knows-where extents.
+However, [[Randall]] dramatically overstates this point here.  After the first step of integration, Randall assumes that any integration can not be solved so simply, and then dives into a step named "????", suggesting that it is unknowable how to proceed.  The rest of the flowchart is (we can assume deliberately) even harder to follow, and does not reach a conclusion.  This is in contrast to the simple, straightforward flowchart for differentiation. The fact that the arrows in the bottom of the integration part lead off the bottom of the page indicates that "Phone calls to mathematicians", "Oh no" and "Burn the evidence" are not final steps in the difficult journey. The flowchart could be extended by Randall to God-knows-where extents.
 It should be noted that Randall slightly undermines his point by providing four different methods, and an "etc", and a "No"-branch for attempting differentiation with no guidelines for selecting between them.

@@ Line 12: / Line 12: @@
 {{w|Derivative|Differentiation}} and {{w|Antiderivative|Integration}} are two major components of {{w|calculus}}. As many Calculus 2 students are painfully aware, integration is much more complicated than the differentiation it undoes.
-However, Randall dramatically overstates this point here.  After the first step of integration, Randall assumes that any integration can not be solved so simply, and then dives into a step named "????", suggesting that it is unknowable how to proceed.  The rest of the flowchart is (we can assume deliberately) even harder to follow, and does not reach a conclusion.  This is in contrast to the simple, straightforward flowchart for differentiation. The fact that the arrows in the bottom of the integration part leads to nowhere indicates that "Phone calls to mathematicians", "Oh no" and "Burn the evidence" are not final steps in the difficult journey. The flowchart could be extended by Randall to God-knows-where extents.
+However, Randall dramatically overstates this point here.  After the first step of integration, Randall assumes that any integration can not be solved so simply, and then dives into a step named "????", suggesting that it is unknowable how to proceed.  The rest of the flowchart is (we can assume deliberately) even harder to follow, and does not reach a conclusion.  This is in contrast to the simple, straightforward flowchart for differentiation. The fact that the arrows in the bottom of the integration part lead off the bottom of the page indicates that "Phone calls to mathematicians", "Oh no" and "Burn the evidence" are not final steps in the difficult journey. The flowchart could be extended by Randall to God-knows-where extents.
 It should be noted that Randall slightly undermines his point by providing four different methods, and an "etc", and a "No"-branch for attempting differentiation with no guidelines for selecting between them.

@@ Line 66: / Line 66: @@
 The Risch algorithm is a notoriously complex procedure that, given a certain class of symbolic integrand, either finds a symbolic integral or proves that no elementary integral exists. (Technically it is only a semi-algorithm, and cannot produce an answer unless it can determine if a certain symbolic expression is {{w|Constant problem|equal to 0}} or not.) Many computer algebra systems have chosen to implement only the simpler Risch-Norman algorithm, which does not come with the same guarantee. A series of extensions to the Risch algorithm extend the class of allowable functions to include (at least) the error function and the logarithmic integral. A human would have to be pretty desperate to attempt this (presumably) by hand.
-'''{{w|Bessel function}}'''
+'''What the heck is a {{w|Bessel function}}??'''
 Bessel functions are the solution to the differential equation x<sup>2</sup> ⋅ d<sup>2</sup>y/dx<sup>2</sup> + x ⋅ dy/dx + (x<sup>2</sup> - n<sup>2</sup>) ⋅ y = 0, where n is the order of the Bessel function. Though such functions show up in some engineering, physics, and abstract mathematics, in lower levels of calculus they are often a sign that the integration was not set up properly before someone put them into a symbolic algebra solver.

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 '''{{w|Bessel function}}'''
-Bessel functions are the solution to the differential equation x<sup>2</sup>⋅d<sup>2</sup>y/dx<sup>2</sup> + x⋅dy/dx + (x<sup>2</sup> - n<sup>2</sup>)⋅y = 0, where n is the order of the Bessel function. Though they do show up in some engineering, physics, and abstract mathematics, in lower levels of calculus they are often a sign that the integration was not set up properly before someone put them into a symbolic algebra solver.
+Bessel functions are the solution to the differential equation x<sup>2</sup> ⋅ d<sup>2</sup>y/dx<sup>2</sup> + x ⋅ dy/dx + (x<sup>2</sup> - n<sup>2</sup>) ⋅ y = 0, where n is the order of the Bessel function. Though such functions show up in some engineering, physics, and abstract mathematics, in lower levels of calculus they are often a sign that the integration was not set up properly before someone put them into a symbolic algebra solver.
 '''Phone calls to mathematicians'''

@@ Line 23: / Line 23: @@
 '''{{w|Power Rule}}'''
-For any 𝑓(𝑥) = 𝑘⋅𝑥<sup>𝑎</sup>, it follows that 𝑓 '(𝑥) = 𝑘⋅𝑎⋅𝑥<sup>𝑎−1</sup>.
+For any 𝑓(𝑥) = 𝑘 ⋅ 𝑥<sup>𝑎</sup>, it follows that 𝑓 '(𝑥) = 𝑘 ⋅ 𝑎 ⋅ 𝑥<sup>𝑎−1</sup>.
 '''{{w|Quotient rule}}'''

@@ Line 40: / Line 40: @@
 '''{{w|Integration by substitution|Substitution}}'''
-The "chain rule" run backwards. Since d(f(u)) = f'(u) ⋅ du/dx ⋅ dx, it follows that f(u) = ∫ f'(u) ⋅ du/dx ⋅ dx. By finding appropriate values for functions f, u such that your problem is in the form ∫ f'(u) ⋅ du/dx ⋅ dx your problem ''may'' be simplified.
+The "chain rule" run backwards. Since d(f(u)) = f'(u) ⋅ du/dx ⋅ dx, it follows that f(u) = ∫ f'(u) ⋅ du/dx ⋅ dx. By finding appropriate values for functions f, u such that your problem is in the form ∫ f'(u) ⋅ du/dx ⋅ dx, your problem ''may'' be simplified.
 '''{{w|Cauchy's integral formula|Cauchy's Formula}}'''