Post History
#6: Post edited
by
TextKit
·
2021-06-02T06:52:58Z (over 3 years ago)
I'm asking merely about definite integrals whose graphs are uneventful, and that high schoolers can effortlessly graph without software, like $\int_0^{\pi/2}\frac{x^2\log^2{(\sin{x})}}{\sin^2x} \, dx$. I used [Desmo](https://www.desmos.com/calculator/gwqjdpjruv) to graph. [u/camelCaseCondition's comment](https://old.reddit.com/r/math/comments/1qpus4/master_of_integration/cdfd2u5/) reaffirms that [the solution](https://math.stackexchange.com/q/1142705) is knotty and effortful, but not why:- >The extent of math that this involves (beyond standard integration techniques usually taught in Calc II, just applied on a large scale), is a significant bit of Complex Analysis (the residue theorem, etc.). In general, everything in his derivation should at least _be understandable_ had you taken Calc I-III and Complex Analysis.
- >However, eyeballing those substitutions and thinking of how to put them all together (and tricks like the mapping from 1/t) to do this is something that probably comes from years of experience using all these techniques and an exceptional cleverness. I can only be in awe when I see the whole thing put together[.]
- 
How can I explain to 16-year-olds, who just started calculus, why it's so nettlesome and tricky to deduce the exact solution by symbolic integration, when the graph looks so unremarkable and straightforward?
- How can I explain to 16-year-olds, who just started calculus, why it's so nettlesome and tricky to symbolically integrate definite integrals, when their graph looks so unremarkable and straightforward?
- I used [Desmo](https://www.desmos.com/calculator/gwqjdpjruv) to graph $\int_0^{\pi/2}\frac{x^2\log^2{(\sin{x})}}{\sin^2x} \, dx$ below. [u/camelCaseCondition's comment](https://old.reddit.com/r/math/comments/1qpus4/master_of_integration/cdfd2u5/) reaffirms that [the solution](https://math.stackexchange.com/q/1142705) is knotty and effortful, but not why:
- >The extent of math that this involves (beyond standard integration techniques usually taught in Calc II, just applied on a large scale), is a significant bit of Complex Analysis (the residue theorem, etc.). In general, everything in his derivation should at least _be understandable_ had you taken Calc I-III and Complex Analysis.
- >However, eyeballing those substitutions and thinking of how to put them all together (and tricks like the mapping from 1/t) to do this is something that probably comes from years of experience using all these techniques and an exceptional cleverness. I can only be in awe when I see the whole thing put together[.]
- 
- My question is more specific than [Why do statements which appear elementary have complicated proofs?](https://math.stackexchange.com/q/5303) and [Why are mathematical proofs so hard?](https://cs.stackexchange.com/q/128980), because I'm asking especially about definite integrals with uneventful graphs, but that high schoolers can effortlessly graph without software.
#5: Post edited
by
TextKit
·
2021-06-02T06:44:25Z (over 3 years ago)
Why can an easily graphable definite integral be knotty to evaluate?
- Why can an easily graphable definite integral, be labyrinthine to evaluate?
I'm asking merely about definite integrals whose graphs are uneventful, and that can effortlessly be graphed high schoolers (without software), like $\int_0^{\pi/2}\frac{x^2\log^2{(\sin{x})}}{\sin^2x} \, dx$. I used [Desmo](https://www.desmos.com/calculator/gwqjdpjruv) to graph. [u/camelCaseCondition's comment](https://old.reddit.com/r/math/comments/1qpus4/master_of_integration/cdfd2u5/) reaffirms that [the solution](https://math.stackexchange.com/q/1142705) is perplexing and effortful, but not why:- >The extent of math that this involves (beyond standard integration techniques usually taught in Calc II, just applied on a large scale), is a significant bit of Complex Analysis (the residue theorem, etc.). In general, everything in his derivation should at least _be understandable_ had you taken Calc I-III and Complex Analysis.
- >However, eyeballing those substitutions and thinking of how to put them all together (and tricks like the mapping from 1/t) to do this is something that probably comes from years of experience using all these techniques and an exceptional cleverness. I can only be in awe when I see the whole thing put together[.]
- 
- How can I explain to 16-year-olds, who just started calculus, why it's so nettlesome and tricky to deduce the exact solution by symbolic integration, when the graph looks so unremarkable and straightforward?
- I'm asking merely about definite integrals whose graphs are uneventful, and that high schoolers can effortlessly graph without software, like $\int_0^{\pi/2}\frac{x^2\log^2{(\sin{x})}}{\sin^2x} \, dx$. I used [Desmo](https://www.desmos.com/calculator/gwqjdpjruv) to graph. [u/camelCaseCondition's comment](https://old.reddit.com/r/math/comments/1qpus4/master_of_integration/cdfd2u5/) reaffirms that [the solution](https://math.stackexchange.com/q/1142705) is knotty and effortful, but not why:
- >The extent of math that this involves (beyond standard integration techniques usually taught in Calc II, just applied on a large scale), is a significant bit of Complex Analysis (the residue theorem, etc.). In general, everything in his derivation should at least _be understandable_ had you taken Calc I-III and Complex Analysis.
- >However, eyeballing those substitutions and thinking of how to put them all together (and tricks like the mapping from 1/t) to do this is something that probably comes from years of experience using all these techniques and an exceptional cleverness. I can only be in awe when I see the whole thing put together[.]
- 
- How can I explain to 16-year-olds, who just started calculus, why it's so nettlesome and tricky to deduce the exact solution by symbolic integration, when the graph looks so unremarkable and straightforward?
#4: Post edited
by
TextKit
·
2021-06-02T06:42:22Z (over 3 years ago)
I'm asking merely about definite integrals whose graphs are uneventful and unremarkable, and that can effortlessly be graphed high schoolers (using software), like $\int_0^{\pi/2}\frac{x^2\log^2{(\sin{x})}}{\sin^2x} \, dx$. I used [Desmo](https://www.desmos.com/calculator/gwqjdpjruv) to graph it. [u/camelCaseCondition's comment](https://old.reddit.com/r/math/comments/1qpus4/master_of_integration/cdfd2u5/) reaffirms that [the solution](https://math.stackexchange.com/q/1142705) is perplexing and effortful, but not why:- >The extent of math that this involves (beyond standard integration techniques usually taught in Calc II, just applied on a large scale), is a significant bit of Complex Analysis (the residue theorem, etc.). In general, everything in his derivation should at least _be understandable_ had you taken Calc I-III and Complex Analysis.
- >However, eyeballing those substitutions and thinking of how to put them all together (and tricks like the mapping from 1/t) to do this is something that probably comes from years of experience using all these techniques and an exceptional cleverness. I can only be in awe when I see the whole thing put together[.]
- I'm asking merely about definite integrals whose graphs are uneventful, and that can effortlessly be graphed high schoolers (without software), like $\int_0^{\pi/2}\frac{x^2\log^2{(\sin{x})}}{\sin^2x} \, dx$. I used [Desmo](https://www.desmos.com/calculator/gwqjdpjruv) to graph. [u/camelCaseCondition's comment](https://old.reddit.com/r/math/comments/1qpus4/master_of_integration/cdfd2u5/) reaffirms that [the solution](https://math.stackexchange.com/q/1142705) is perplexing and effortful, but not why:
- >The extent of math that this involves (beyond standard integration techniques usually taught in Calc II, just applied on a large scale), is a significant bit of Complex Analysis (the residue theorem, etc.). In general, everything in his derivation should at least _be understandable_ had you taken Calc I-III and Complex Analysis.
- >However, eyeballing those substitutions and thinking of how to put them all together (and tricks like the mapping from 1/t) to do this is something that probably comes from years of experience using all these techniques and an exceptional cleverness. I can only be in awe when I see the whole thing put together[.]
- 
- How can I explain to 16-year-olds, who just started calculus, why it's so nettlesome and tricky to deduce the exact solution by symbolic integration, when the graph looks so unremarkable and straightforward?
#3: Post edited
by
TextKit
·
2021-06-02T06:38:41Z (over 3 years ago)
High-school students can sketch $\int_0^{\pi/2}\frac{x^2\log^2{(\sin{x})}}{\sin^2x} \,dx$, knowing only univariate calculus (I used [Desmo](https://www.desmos.com/calculator/gwqjdpjruv)). Though its graph below appears straightforward, why's [the solution](https://math.stackexchange.com/q/1142705) so perplexing and effortful as acknowledged by [u/camelCaseCondition](https://old.reddit.com/r/math/comments/1qpus4/master_of_integration/cdfd2u5/)?- >The extent of math that this involves (beyond standard integration techniques usually taught in Calc II, just applied on a large scale), is a significant bit of Complex Analysis (the residue theorem, etc.). In general, everything in his derivation should at least _be understandable_ had you taken Calc I-III and Complex Analysis.
- >However, eyeballing those substitutions and thinking of how to put them all together (and tricks like the mapping from 1/t) to do this is something that probably comes from years of experience using all these techniques and an exceptional cleverness. I can only be in awe when I see the whole thing put together[.]
- 
- I'm asking merely about definite integrals whose graphs are uneventful and unremarkable, and that can effortlessly be graphed high schoolers (using software), like $\int_0^{\pi/2}\frac{x^2\log^2{(\sin{x})}}{\sin^2x} \, dx$. I used [Desmo](https://www.desmos.com/calculator/gwqjdpjruv) to graph it. [u/camelCaseCondition's comment](https://old.reddit.com/r/math/comments/1qpus4/master_of_integration/cdfd2u5/) reaffirms that [the solution](https://math.stackexchange.com/q/1142705) is perplexing and effortful, but not why:
- >The extent of math that this involves (beyond standard integration techniques usually taught in Calc II, just applied on a large scale), is a significant bit of Complex Analysis (the residue theorem, etc.). In general, everything in his derivation should at least _be understandable_ had you taken Calc I-III and Complex Analysis.
- >However, eyeballing those substitutions and thinking of how to put them all together (and tricks like the mapping from 1/t) to do this is something that probably comes from years of experience using all these techniques and an exceptional cleverness. I can only be in awe when I see the whole thing put together[.]
- 
#2: Post edited
by
TextKit
·
2021-06-02T06:31:59Z (over 3 years ago)
High-school students can sketch $\int_0^{\pi/2}\frac{x^2\log^2{(\sin{x})}}{\sin^2x} \,dx$, knowing only univariate calculus (I used [Desmo](https://www.desmos.com/calculator/gwqjdpjruv)). Though the graph appears straightforward, why's [the solution](https://math.stackexchange.com/q/1142705) so knotty as acknowledged by [u/camelCaseCondition](https://old.reddit.com/r/math/comments/1qpus4/master_of_integration/cdfd2u5/)?- >The extent of math that this involves (beyond standard integration techniques usually taught in Calc II, just applied on a large scale), is a significant bit of Complex Analysis (the residue theorem, etc.). In general, everything in his derivation should at least _be understandable_ had you taken Calc I-III and Complex Analysis.
- >However, eyeballing those substitutions and thinking of how to put them all together (and tricks like the mapping from 1/t) to do this is something that probably comes from years of experience using all these techniques and an exceptional cleverness. I can only be in awe when I see the whole thing put together[.]
- 
- High-school students can sketch $\int_0^{\pi/2}\frac{x^2\log^2{(\sin{x})}}{\sin^2x} \,dx$, knowing only univariate calculus (I used [Desmo](https://www.desmos.com/calculator/gwqjdpjruv)). Though its graph below appears straightforward, why's [the solution](https://math.stackexchange.com/q/1142705) so perplexing and effortful as acknowledged by [u/camelCaseCondition](https://old.reddit.com/r/math/comments/1qpus4/master_of_integration/cdfd2u5/)?
- >The extent of math that this involves (beyond standard integration techniques usually taught in Calc II, just applied on a large scale), is a significant bit of Complex Analysis (the residue theorem, etc.). In general, everything in his derivation should at least _be understandable_ had you taken Calc I-III and Complex Analysis.
- >However, eyeballing those substitutions and thinking of how to put them all together (and tricks like the mapping from 1/t) to do this is something that probably comes from years of experience using all these techniques and an exceptional cleverness. I can only be in awe when I see the whole thing put together[.]
- 
#1: Initial revision
by
TextKit
·
2021-06-02T06:29:00Z (over 3 years ago)
Why can an easily graphable definite integral be knotty to evaluate?
High-school students can sketch $\int_0^{\pi/2}\frac{x^2\log^2{(\sin{x})}}{\sin^2x} \,dx$, knowing only univariate calculus (I used [Desmo](https://www.desmos.com/calculator/gwqjdpjruv)). Though the graph appears straightforward, why's [the solution](https://math.stackexchange.com/q/1142705) so knotty as acknowledged by [u/camelCaseCondition](https://old.reddit.com/r/math/comments/1qpus4/master_of_integration/cdfd2u5/)?
>The extent of math that this involves (beyond standard integration techniques usually taught in Calc II, just applied on a large scale), is a significant bit of Complex Analysis (the residue theorem, etc.). In general, everything in his derivation should at least _be understandable_ had you taken Calc I-III and Complex Analysis.
>However, eyeballing those substitutions and thinking of how to put them all together (and tricks like the mapping from 1/t) to do this is something that probably comes from years of experience using all these techniques and an exceptional cleverness. I can only be in awe when I see the whole thing put together[.]
