[math-fun] Did any of you integrate √tan in freshman calculus?
In[38]:= Integrate[√Tan[x], x] Out[38]= 2/3 Hypergeometric2F1[3/4, 1, 7/4, -Tan[x]^2] Tan[x]^(3/2) In[39]:= FunctionExpand@% Out[39]= 2/3 Tan[x]^( 3/2) (-((3 ArcTan[(-Tan[x]^2)^(1/4)])/(2 (-Tan[x]^2)^(3/4))) + ( 3 ArcTanh[(-Tan[x]^2)^(1/4)])/(2 (-Tan[x]^2)^(3/4))) I'm pretty sure this would've left me some PTSD. And again later with Macsyma's Risch implementation. I suspect a conspiracy to hide this from emotionally vulnerable undergrads. —rwg
Bill, That was marked 'only for the most ambitious' on a sheet of 66 practice integrals for first-year undergrads: http://www.damtp.cam.ac.uk/user/examples/A3La.pdf -- APG.
Sent: Saturday, June 22, 2019 at 2:54 AM From: "Bill Gosper" <billgosper@gmail.com> To: math-fun@mailman.xmission.com Subject: [math-fun] Did any of you integrate √tan in freshman calculus?
In[38]:= Integrate[√Tan[x], x]
Out[38]= 2/3 Hypergeometric2F1[3/4, 1, 7/4, -Tan[x]^2] Tan[x]^(3/2)
In[39]:= FunctionExpand@%
Out[39]= 2/3 Tan[x]^( 3/2) (-((3 ArcTan[(-Tan[x]^2)^(1/4)])/(2 (-Tan[x]^2)^(3/4))) + ( 3 ArcTanh[(-Tan[x]^2)^(1/4)])/(2 (-Tan[x]^2)^(3/4)))
I'm pretty sure this would've left me some PTSD. And again later with Macsyma's Risch implementation. I suspect a conspiracy to hide this from emotionally vulnerable undergrads. —rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Hah! I like how the practice sheet copped out on the answer for that one: 66) A little too cumbersome to present here! Compare notes with a friend. Tom Adam P. Goucher writes:
Bill,
That was marked 'only for the most ambitious' on a sheet of 66 practice integrals for first-year undergrads:
http://www.damtp.cam.ac.uk/user/examples/A3La.pdf
-- APG.
Sent: Saturday, June 22, 2019 at 2:54 AM From: "Bill Gosper" <billgosper@gmail.com> To: math-fun@mailman.xmission.com Subject: [math-fun] Did any of you integrate √tan in freshman calculus?
In[38]:= Integrate[√Tan[x], x]
Out[38]= 2/3 Hypergeometric2F1[3/4, 1, 7/4, -Tan[x]^2] Tan[x]^(3/2)
In[39]:= FunctionExpand@%
Out[39]= 2/3 Tan[x]^( 3/2) (-((3 ArcTan[(-Tan[x]^2)^(1/4)])/(2 (-Tan[x]^2)^(3/4))) + ( 3 ArcTanh[(-Tan[x]^2)^(1/4)])/(2 (-Tan[x]^2)^(3/4)))
I'm pretty sure this would've left me some PTSD. And again later with Macsyma's Risch implementation. I suspect a conspiracy to hide this from emotionally vulnerable undergrads. —rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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I'd first substitute x = arctan y, so the integral becomes int sqrt(y)/(1+y^2) dy, and then let y = z^2, so the integral is int 2 * z^2/(1+ z^4) dz, and then use partial fractions. On Sat, Jun 22, 2019 at 6:12 AM Tom Karzes <karzes@sonic.net> wrote:
Hah! I like how the practice sheet copped out on the answer for that one:
66) A little too cumbersome to present here! Compare notes with a friend.
Tom
Adam P. Goucher writes:
Bill,
That was marked 'only for the most ambitious' on a sheet of 66 practice integrals for first-year undergrads:
http://www.damtp.cam.ac.uk/user/examples/A3La.pdf
-- APG.
Sent: Saturday, June 22, 2019 at 2:54 AM From: "Bill Gosper" <billgosper@gmail.com> To: math-fun@mailman.xmission.com Subject: [math-fun] Did any of you integrate √tan in freshman calculus?
In[38]:= Integrate[√Tan[x], x]
Out[38]= 2/3 Hypergeometric2F1[3/4, 1, 7/4, -Tan[x]^2] Tan[x]^(3/2)
In[39]:= FunctionExpand@%
Out[39]= 2/3 Tan[x]^( 3/2) (-((3 ArcTan[(-Tan[x]^2)^(1/4)])/(2 (-Tan[x]^2)^(3/4))) + ( 3 ArcTanh[(-Tan[x]^2)^(1/4)])/(2 (-Tan[x]^2)^(3/4)))
I'm pretty sure this would've left me some PTSD. And again later with Macsyma's Risch implementation. I suspect a conspiracy to hide this from emotionally vulnerable undergrads. —rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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On 2019-06-22 05:47, Victor Miller wrote:
I'd first substitute x = arctan y, so the integral becomes int sqrt(y)/(1+y^2) dy, and then let y = z^2, so the integral is int 2 * z^2/(1+ z^4) dz, and then use partial fractions.
In[34]:= FullSimplify@ D[√I (ArcTan[√(I Tan[x])] - ArcTanh[√(I Tan[x])]), x] // PowerExpand Out[34]= Sqrt[Tan[x]] —rwg
On Sat, Jun 22, 2019 at 6:12 AM Tom Karzes <karzes@sonic.net> wrote:
Hah! I like how the practice sheet copped out on the answer for that one:
66) A little too cumbersome to present here! Compare notes with a friend.
Tom
Adam P. Goucher writes:
Bill,
That was marked 'only for the most ambitious' on a sheet of 66 practice integrals for first-year undergrads:
http://www.damtp.cam.ac.uk/user/examples/A3La.pdf
-- APG.
Sent: Saturday, June 22, 2019 at 2:54 AM From: "Bill Gosper" <billgosper@gmail.com> To: math-fun@mailman.xmission.com Subject: [math-fun] Did any of you integrate √tan in freshman calculus?
In[38]:= Integrate[√Tan[x], x]
Out[38]= 2/3 Hypergeometric2F1[3/4, 1, 7/4, -Tan[x]^2] Tan[x]^(3/2)
In[39]:= FunctionExpand@%
Out[39]= 2/3 Tan[x]^( 3/2) (-((3 ArcTan[(-Tan[x]^2)^(1/4)])/(2 (-Tan[x]^2)^(3/4))) + ( 3 ArcTanh[(-Tan[x]^2)^(1/4)])/(2 (-Tan[x]^2)^(3/4)))
I'm pretty sure this would've left me some PTSD. And again later with Macsyma's Risch implementation. I suspect a conspiracy to hide this from emotionally vulnerable undergrads. —rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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By differentiation under the integral sign, we can also find F(y)=sqrt(y)/(1+y^2) a D-finite function, and then: Factor@Dot[{3*y^2 - 1 , 2*y^3 + 2*y}, D[Sqrt[y]/(1 + y^2), {y, #}] & /@ {0, 1}] == 0 This identity should lead quickly to the coefficient recursion. --Brad On Sat, Jun 22, 2019 at 7:48 AM Victor Miller <victorsmiller@gmail.com> wrote:
I'd first substitute x = arctan y, so the integral becomes int sqrt(y)/(1+y^2) dy, and then let y = z^2, so the integral is int 2 * z^2/(1+ z^4) dz, and then use partial fractions.
On Sat, Jun 22, 2019 at 6:12 AM Tom Karzes <karzes@sonic.net> wrote:
Hah! I like how the practice sheet copped out on the answer for that one:
66) A little too cumbersome to present here! Compare notes with a friend.
Tom
Adam P. Goucher writes:
Bill,
That was marked 'only for the most ambitious' on a sheet of 66 practice integrals for first-year undergrads:
http://www.damtp.cam.ac.uk/user/examples/A3La.pdf
-- APG.
Sent: Saturday, June 22, 2019 at 2:54 AM From: "Bill Gosper" <billgosper@gmail.com> To: math-fun@mailman.xmission.com Subject: [math-fun] Did any of you integrate √tan in freshman calculus?
In[38]:= Integrate[√Tan[x], x]
Out[38]= 2/3 Hypergeometric2F1[3/4, 1, 7/4, -Tan[x]^2] Tan[x]^(3/2)
In[39]:= FunctionExpand@%
Out[39]= 2/3 Tan[x]^( 3/2) (-((3 ArcTan[(-Tan[x]^2)^(1/4)])/(2 (-Tan[x]^2)^(3/4))) + ( 3 ArcTanh[(-Tan[x]^2)^(1/4)])/(2 (-Tan[x]^2)^(3/4)))
I'm pretty sure this would've left me some PTSD. And again later with Macsyma's Risch implementation. I suspect a conspiracy to hide this from emotionally vulnerable undergrads. —rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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I seem to recall a result in Watson's Bessel function Treatise that he deemed "too cumbrous to be of any importance". Fortunately, the same wasn't said of Maxwell's Equations in their original form.-) —rwg (I just verified that PC Macsyma integrates √tan.) On 2019-06-22 03:11, Tom Karzes wrote:
Hah! I like how the practice sheet copped out on the answer for that one:
66) A little too cumbersome to present here! Compare notes with a friend.
Tom
Adam P. Goucher writes:
Bill,
That was marked 'only for the most ambitious' on a sheet of 66 practice integrals for first-year undergrads:
http://www.damtp.cam.ac.uk/user/examples/A3La.pdf
-- APG.
Sent: Saturday, June 22, 2019 at 2:54 AM From: "Bill Gosper" <billgosper@gmail.com> To: math-fun@mailman.xmission.com Subject: [math-fun] Did any of you integrate √tan in freshman calculus?
In[38]:= Integrate[√Tan[x], x]
Out[38]= 2/3 Hypergeometric2F1[3/4, 1, 7/4, -Tan[x]^2] Tan[x]^(3/2)
In[39]:= FunctionExpand@%
Out[39]= 2/3 Tan[x]^( 3/2) (-((3 ArcTan[(-Tan[x]^2)^(1/4)])/(2 (-Tan[x]^2)^(3/4))) + ( 3 ArcTanh[(-Tan[x]^2)^(1/4)])/(2 (-Tan[x]^2)^(3/4)))
I'm pretty sure this would've left me some PTSD. And again later with Macsyma's Risch implementation. I suspect a conspiracy to hide this from emotionally vulnerable undergrads. —rwg _______________________________________________
Maple also gives a solution in terms of cos, arccos, sin, sqrt(tan) etc: sqrt(tan(x))*cos(x)*sqrt(2)*arccos(cos(x) - sin(x))/(2*sqrt(cos(x)*sin(x))) - sqrt(2)*ln(cos(x) + sqrt(2)*sqrt(tan(x))*cos(x) + sin(x))/2 Steve
On Jun 22, 2019, at 11:05 AM, rwg <rwg@ma.sdf.org> wrote:
I seem to recall a result in Watson's Bessel function Treatise that he deemed "too cumbrous to be of any importance". Fortunately, the same wasn't said of Maxwell's Equations in their original form.-) —rwg (I just verified that PC Macsyma integrates √tan.)
On 2019-06-22 03:11, Tom Karzes wrote:
Hah! I like how the practice sheet copped out on the answer for that one: 66) A little too cumbersome to present here! Compare notes with a friend. Tom Adam P. Goucher writes:
Bill,
That was marked 'only for the most ambitious' on a sheet of 66 practice integrals for first-year undergrads:
https://urldefense.proofpoint.com/v2/url?u=http-3A__www.damtp.cam.ac.uk_user...
-- APG.
Sent: Saturday, June 22, 2019 at 2:54 AM From: "Bill Gosper" <billgosper@gmail.com> To: math-fun@mailman.xmission.com Subject: [math-fun] Did any of you integrate √tan in freshman calculus?
In[38]:= Integrate[√Tan[x], x]
Out[38]= 2/3 Hypergeometric2F1[3/4, 1, 7/4, -Tan[x]^2] Tan[x]^(3/2)
In[39]:= FunctionExpand@%
Out[39]= 2/3 Tan[x]^( 3/2) (-((3 ArcTan[(-Tan[x]^2)^(1/4)])/(2 (-Tan[x]^2)^(3/4))) + ( 3 ArcTanh[(-Tan[x]^2)^(1/4)])/(2 (-Tan[x]^2)^(3/4)))
I'm pretty sure this would've left me some PTSD. And again later with Macsyma's Risch implementation. I suspect a conspiracy to hide this from emotionally vulnerable undergrads. —rwg _______________________________________________
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://urldefense.proofpoint.com/v2/url?u=https-3A__mailman.xmission.com_cg...
participants (7)
-
Adam P. Goucher -
Bill Gosper -
Brad Klee -
Lucas, Stephen K - lucassk -
rwg -
Tom Karzes -
Victor Miller