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Title text: "Symbolic integration" is when you theatrically go through the motions of finding integrals, but the actual result you get doesn't matter because it's purely symbolic.

Explanation

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This comic provides a flowchart purporting to show the process of differentiation, and another for integration.

Differentiation and Integration are two major components of 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-know-where extents.

It should be noted that Randall slightly undermines his point by providing four different methods, and an "etc", for attempting differentiation with no guidelines for selecting between them.

Differentiation

Chain rule

For any and , it follows that .

Power Rule

For any , it follows that .

Quotient rule

For any and , it follows that if .

Product rule

For any and , it follows that .

Integration

Integration by parts

The "product rule" run backwards. Since , it follows that by integrating both sides you get , which is more commonly written as . By finding appropriate values for functions such that your problem is in the form , your problem may be simplified.

Substitution

The "chain rule" run backwards. Since , it follows that . By finding appropriate values for functions such that your problem is in the form your problem may be simplified.

Cauchy's Formula

Cauchy's Integral formula is a result in complex analysis that relates the value of a contour integral in the complex plane to properties of the singularities in the interior of the contour. It is often used to compute integrals on the real line by extending the path of the integral from the real line into the complex plane to apply the formula, then proving that the integral from the parts of the contour not on the real line has value zero.

Partial Fractions

Partial fractions is a technique for breaking up a function that comprises one polynomial divided by another into a sum of functions comprising constants over the factors of the original denominator, which can easily be integrated into logarithms.

Install Mathematica

Mathematica is a modern technical computing system spanning most areas. One of its features is to compute mathematical functions. This step in the flowchart is install and use Mathematica to compute to do the integration for you.

Riemann Integration The Riemann integral is a definition of definite integration. Elementary textbooks on calculus sometimes present finding a definite integral as a process of approximating an area by strips of equal width and then taking the limit as the strips become narrower. Riemann integration removes the requirement that the strips have equal width, and so is a more flexible definition. However there are still many functions for which the Riemann integral doesn't converge, and consideration of these functions leads to the Lesbegue Integral. Riemann integration is not a method of calculus appropriate for finding the anti-derivative of an elementary function.

Stokes' Theorem ???

Risch Algorithm The Risch Algorithm is a complex procedure that reduces the process of symbolic integration to purely algebraic steps. It is implemented in Computer Algebra software, such as Mathematica.

Bessel function ???

Symbolic integration Mentioned in the title text. ???

Burn the evidence In a parody of detective novels, this may refer to the burning of your work to avoid the shame of being unable to solve the integration problem. Alternatively, this could be an ironic hint to the fact that in order to find the integral, it may even be necessary to break the law or upset higher powers, so that the negative consequences of a persecution can only be avoided by destroying the evidence.

Transcript

This transcript is incomplete. Please help editing it! Thanks.

[Two flow charts are shown.]

[The first flow chart has four steps in simple order, one with multiple recommendations.] DIFFERENTIATION Start Try applying Chain Rule Power Rule Quotient Rule Product Rule Etc. Done? Yes No Done!

[The second flow chart begins like the first, then descends into chaos.]

INTEGRATION Start Try applying Integration by Parts Substitution Done? Haha, Nope!

[Chaos, Roughly from left to right, top to bottom, direction arrows not included.] Cauchy's Formula ???? ???!? ??? ??? ? Partial Fractions ?? ? Install Mathematica ? Riemann Integration Stokes' Theorem ??? ? Risch Algorithm ??? [Sad face.] ????? ??? What the heck is a Bessel Function?? Phone calls to mathematicians Oh No Burn the Evidence



