Great Question! The answer depends on what integrals you refer to. Not every elementary function (i.e. those from the video) has an elementary antiderivative (e.g. exp(-x^2) ), so a Fundamental Theorem of Calculus route would be a problem. For definite integrals, it depends on the construction. For example, Lebesgue integrals are built using "simple functions" which clarify the key ideas.
@@artkalbphdI’m not really well versed on Lebesgue integrals, so I’m not sure if they would be needed for a minimum amount of integrals. But in the case of e^(-x²), one could use the erf(x), though that would mean that one would also have to include such functions for thing like the logarithmic function or sinc(x)… Although with the five rules in the video it wouldn’t be possible to say that |x|’ = sgn(x) or |x|/x or differentiate the Weierstrass function, so one could also leave nonelementary integrals out of the question. If that were the case, how many would you need?
@@aiyajeezyx6498 the derivatives |x| (as a piecewise function) and Weierstrass function (as a series) could be found primarily using the rules plus an analysis of limits, which do fail to exist at the corner of |x| and everywhere in Weierstrass. For integration of elementary functions, I would guess 1, linearity, u sub, exp, sin as inverses of The Big Five, plus integration by parts and power rule to get most of them. Harder to say because you quickly hit nonelementary functions, but you don't immediately know for sure.
@@artkalbphdwouldnt you be able to remove integration by parts cuz of the product rule? Cuz int((uv)') = int(udv) + int(vdu)? And 1 can be derived from the power rule. Maybe ln instead?
nice compact and useful video
This is a very nice and compact overview. Thanks for sharing this!
Oh my Euler, oh my Euler!!! I just found a great channel!!! I'm so happy right now!!!
314th view 👍
This is really cool! But would it be possible to make a similar set of minimal rules needed for integrals?
Great Question! The answer depends on what integrals you refer to.
Not every elementary function (i.e. those from the video) has an elementary antiderivative (e.g. exp(-x^2) ), so a Fundamental Theorem of Calculus route would be a problem.
For definite integrals, it depends on the construction. For example, Lebesgue integrals are built using "simple functions" which clarify the key ideas.
@@artkalbphdI’m not really well versed on Lebesgue integrals, so I’m not sure if they would be needed for a minimum amount of integrals. But in the case of e^(-x²), one could use the erf(x), though that would mean that one would also have to include such functions for thing like the logarithmic function or sinc(x)…
Although with the five rules in the video it wouldn’t be possible to say that |x|’ = sgn(x) or |x|/x or differentiate the Weierstrass function, so one could also leave nonelementary integrals out of the question. If that were the case, how many would you need?
@@aiyajeezyx6498 the derivatives |x| (as a piecewise function) and Weierstrass function (as a series) could be found primarily using the rules plus an analysis of limits, which do fail to exist at the corner of |x| and everywhere in Weierstrass.
For integration of elementary functions, I would guess 1, linearity, u sub, exp, sin as inverses of The Big Five, plus integration by parts and power rule to get most of them. Harder to say because you quickly hit nonelementary functions, but you don't immediately know for sure.
@@artkalbphdwouldnt you be able to remove integration by parts cuz of the product rule? Cuz int((uv)') = int(udv) + int(vdu)? And 1 can be derived from the power rule. Maybe ln instead?
@@R0mm3n both good points