Re: Do some definite integral calculation.
- To: mathgroup at smc.vnet.net
- Subject: [mg99486] Re: [mg99393] Do some definite integral calculation.
- From: hongyi.zhao at gmail.com
- Date: Wed, 6 May 2009 05:28:37 -0400 (EDT)
- References: <200905050938.FAA20515@smc.vnet.net> <op.utgvq8zotgfoz2@bobbys-imac.local>
On Wednesday, May 6, 2009 at 1:19, btreat1 at austin.rr.com wrote: > Your post is unreadable, but guesswork leads me to: > Clear[a, b, c, x, theta] > Off[Solve::"ifun"] > tanTheta = > t /. Last@ > Solve[{t == Tan[theta], Sin[theta] == a x^2 + b x + c}, t, > theta] // Simplify > Sqrt[-(c + x (b + a x))^2]/Sqrt[-1 + c^2 + b^2 x^2 + 2 a b x^3 + > a^2 x^4 + 2 c x (b + a x)] Thanks a lot, you're absolutely right. > (First could be used, rather than Last.) What do mean by saying this? > y=Integrate[tanTheta, x] > (2 c ((-b - Sqrt[-4 a + b^2 - 4 a c])/( > 2 a) - (-b + Sqrt[4 a + b^2 - 4 a c])/( > 2 a)) Sqrt[((-((-b + Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + (-b + > Sqrt[4 a + b^2 - 4 a c])/( > 2 a)) (-((-b - Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + > x))/((-((-b - Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + (-b + Sqrt[ > 4 a + b^2 - 4 a c])/( > 2 a)) (-((-b + Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + > x))] (-((-b + Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + > x)^2 Sqrt[((-((-b - Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + (-b + > Sqrt[-4 a + b^2 - 4 a c])/( > 2 a)) (-((-b - Sqrt[4 a + b^2 - 4 a c])/(2 a)) + > x))/((-((-b - Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + (-b - Sqrt[ > 4 a + b^2 - 4 a c])/( > 2 a)) (-((-b + Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + x))] > Sqrt[((-((-b - Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + (-b + > Sqrt[-4 a + b^2 - 4 a c])/( > 2 a)) (-((-b + Sqrt[4 a + b^2 - 4 a c])/(2 a)) + > x))/((-((-b - Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + (-b + Sqrt[ > 4 a + b^2 - 4 a c])/( > 2 a)) (-((-b + Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + x))] > Sqrt[-1 + c^2 + 2 b c x + b^2 x^2 + 2 a c x^2 + 2 a b x^3 + > a^2 x^4] Sqrt[-(c + x (b + a x))^2] > EllipticF[ > ArcSin[Sqrt[((-((-b + Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + (-b + > Sqrt[4 a + b^2 - 4 a c])/( > 2 a)) (-((-b - Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + > x))/((-((-b - Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + (-b + Sqrt[ > 4 a + b^2 - 4 a c])/( > 2 a)) (-((-b + Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + > x))]], (((-b + Sqrt[-4 a + b^2 - 4 a c])/( > 2 a) - (-b - Sqrt[4 a + b^2 - 4 a c])/( > 2 a)) ((-b - Sqrt[-4 a + b^2 - 4 a c])/( > 2 a) - (-b + Sqrt[4 a + b^2 - 4 a c])/( > 2 a)))/(((-b - Sqrt[-4 a + b^2 - 4 a c])/( > 2 a) - (-b - Sqrt[4 a + b^2 - 4 a c])/( > 2 a)) ((-b + Sqrt[-4 a + b^2 - 4 a c])/( > 2 a) - (-b + Sqrt[4 a + b^2 - 4 a c])/(2 a)))])/((-((-b - > Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + (-b + > Sqrt[-4 a + b^2 - 4 a c])/( > 2 a)) (-((-b + Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + (-b + Sqrt[ > 4 a + b^2 - 4 a c])/(2 a)) Sqrt[-1 + c^2 + > 2 b c x + (b^2 + 2 a c) x^2 + 2 a b x^3 + > a^2 x^4] (c + x (b + a x)) Sqrt[-1 + c^2 + b^2 x^2 + 2 a b x^3 + > a^2 x^4 + > 2 c x (b + a x)]) + (2 b ((-b - Sqrt[-4 a + b^2 - 4 a c])/( > 2 a) - (-b + Sqrt[4 a + b^2 - 4 a c])/( > 2 a)) (-((-b + Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + x)^2 Sqrt[( > Sqrt[-4 a + b^2 - > 4 a c] (-((-b - Sqrt[4 a + b^2 - 4 a c])/(2 a)) + x))/( > a (-((-b - Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + (-b - Sqrt[ > 4 a + b^2 - 4 a c])/( > 2 a)) (-((-b + Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + x))] Sqrt[( > Sqrt[-4 a + b^2 - > 4 a c] (-((-b + Sqrt[4 a + b^2 - 4 a c])/(2 a)) + x))/( > a (-((-b - Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + (-b + Sqrt[ > 4 a + b^2 - 4 a c])/( > 2 a)) (-((-b + Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + x))] > Sqrt[((Sqrt[-4 a + b^2 - 4 a c] - Sqrt[4 a + b^2 - 4 a c]) (b + > Sqrt[-4 a + b^2 - 4 a c] + 2 a x))/((Sqrt[-4 a + b^2 - 4 a c] + > Sqrt[4 a + b^2 - 4 a c]) (-b + Sqrt[-4 a + b^2 - 4 a c] - > 2 a x))] > Sqrt[-1 + c^2 + 2 b c x + b^2 x^2 + 2 a c x^2 + 2 a b x^3 + > a^2 x^4] Sqrt[-(c + x (b + a x))^2] (-(1/( > 2 a))(-b + Sqrt[-4 a + b^2 - 4 a c]) EllipticF[ > ArcSin[Sqrt[((Sqrt[-4 a + b^2 - 4 a c] - Sqrt[ > 4 a + b^2 - 4 a c]) (b + Sqrt[-4 a + b^2 - 4 a c] + > 2 a x))/((Sqrt[-4 a + b^2 - 4 a c] + Sqrt[ > 4 a + b^2 - 4 a c]) (-b + Sqrt[-4 a + b^2 - 4 a c] - > 2 a x))]], (Sqrt[-4 a + b^2 - 4 a c] + Sqrt[ > 4 a + b^2 - 4 a c])^2/(Sqrt[-4 a + b^2 - 4 a c] - Sqrt[ > 4 a + b^2 - 4 a c])^2] + (1/a) > Sqrt[-4 a + b^2 - 4 a c] > EllipticPi[(-((-b - Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + (-b + > Sqrt[4 a + b^2 - 4 a c])/( > 2 a))/(-((-b + Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + (-b + Sqrt[ > 4 a + b^2 - 4 a c])/(2 a)), > ArcSin[Sqrt[((Sqrt[-4 a + b^2 - 4 a c] - Sqrt[ > 4 a + b^2 - 4 a c]) (b + Sqrt[-4 a + b^2 - 4 a c] + > 2 a x))/((Sqrt[-4 a + b^2 - 4 a c] + Sqrt[ > 4 a + b^2 - 4 a c]) (-b + Sqrt[-4 a + b^2 - 4 a c] - > 2 a x))]], (Sqrt[-4 a + b^2 - 4 a c] + Sqrt[ > 4 a + b^2 - 4 a c])^2/(Sqrt[-4 a + b^2 - 4 a c] - Sqrt[ > 4 a + b^2 - 4 a c])^2]))/((-((-b - > Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + (-b + > Sqrt[-4 a + b^2 - 4 a c])/( > 2 a)) ((-b + Sqrt[-4 a + b^2 - 4 a c])/( > 2 a) - (-b + Sqrt[4 a + b^2 - 4 a c])/(2 a)) Sqrt[-1 + c^2 + > 2 b c x + (b^2 + 2 a c) x^2 + 2 a b x^3 + > a^2 x^4] (c + x (b + a x)) Sqrt[-1 + c^2 + b^2 x^2 + 2 a b x^3 + > a^2 x^4 + 2 c x (b + a x)]) + (a Sqrt[-1 + c^2 + 2 b c x + > b^2 x^2 + 2 a c x^2 + 2 a b x^3 + a^2 x^4] > Sqrt[-(c + > x (b + a x))^2] ((-((-b - Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + > x) (-((-b - Sqrt[4 a + b^2 - 4 a c])/(2 a)) + > x) (-((-b + Sqrt[4 a + b^2 - 4 a c])/(2 a)) + > x) + (-((-b - Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + (-b + Sqrt[ > 4 a + b^2 - 4 a c])/( > 2 a)) (-((-b + Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + x)^2 Sqrt[( > Sqrt[-4 a + b^2 - > 4 a c] (-((-b - Sqrt[4 a + b^2 - 4 a c])/(2 a)) + x))/( > a (-((-b - Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + (-b - Sqrt[ > 4 a + b^2 - 4 a c])/( > 2 a)) (-((-b + Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + x))] > Sqrt[(Sqrt[-4 a + b^2 - > 4 a c] (-((-b + Sqrt[4 a + b^2 - 4 a c])/(2 a)) + x))/( > a (-((-b - Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + (-b + Sqrt[ > 4 a + b^2 - 4 a c])/( > 2 a)) (-((-b + Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + x))] > Sqrt[((Sqrt[-4 a + b^2 - 4 a c] - Sqrt[ > 4 a + b^2 - 4 a c]) (b + Sqrt[-4 a + b^2 - 4 a c] + > 2 a x))/((Sqrt[-4 a + b^2 - 4 a c] + Sqrt[ > 4 a + b^2 - 4 a c]) (-b + Sqrt[-4 a + b^2 - 4 a c] - > 2 a x))] ((1/Sqrt[-4 a + b^2 - 4 a c]) > a (-((-b - Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + (-b - Sqrt[ > 4 a + b^2 - 4 a c])/(2 a)) EllipticE[ > ArcSin[Sqrt[((Sqrt[-4 a + b^2 - 4 a c] - Sqrt[ > 4 a + b^2 - 4 a c]) (b + Sqrt[-4 a + b^2 - 4 a c] + > 2 a x))/((Sqrt[-4 a + b^2 - 4 a c] + Sqrt[ > 4 a + b^2 - 4 a c]) (-b + Sqrt[-4 a + b^2 - 4 a c] - > 2 a x))]], (Sqrt[-4 a + b^2 - 4 a c] + Sqrt[ > 4 a + b^2 - 4 a c])^2/(Sqrt[-4 a + b^2 - 4 a c] - Sqrt[ > 4 a + b^2 - > 4 a c])^2] + (a (((-b + > Sqrt[-4 a + b^2 - > 4 a c]) (-((-b + Sqrt[-4 a + b^2 - 4 a c])/( > 2 a)) - (-b + Sqrt[4 a + b^2 - 4 a c])/(2 a)))/( > 2 a) - ((-b - > Sqrt[-4 a + b^2 - 4 a c]) ((-b + > Sqrt[-4 a + b^2 - 4 a c])/( > 2 a) - (-b + Sqrt[4 a + b^2 - 4 a c])/(2 a)))/( > 2 a)) EllipticF[ > ArcSin[Sqrt[((Sqrt[-4 a + b^2 - 4 a c] - Sqrt[ > 4 a + b^2 - 4 a c]) (b + Sqrt[-4 a + b^2 - 4 a c] + > 2 a x))/((Sqrt[-4 a + b^2 - 4 a c] + Sqrt[ > 4 a + b^2 - 4 a c]) (-b + Sqrt[-4 a + b^2 - 4 a c] - > 2 a x))]], (Sqrt[-4 a + b^2 - 4 a c] + Sqrt[ > 4 a + b^2 - 4 a c])^2/(Sqrt[-4 a + b^2 - 4 a c] - Sqrt[ > 4 a + b^2 - 4 a c])^2])/(Sqrt[-4 a + b^2 - > 4 a c] (-((-b + Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + (-b + > Sqrt[4 a + b^2 - 4 a c])/( > 2 a))) - ((-((-b - Sqrt[-4 a + b^2 - 4 a c])/( > 2 a)) - (-b + Sqrt[-4 a + b^2 - 4 a c])/( > 2 a) - (-b - Sqrt[4 a + b^2 - 4 a c])/( > 2 a) - (-b + Sqrt[4 a + b^2 - 4 a c])/( > 2 a)) EllipticPi[(-((-b - Sqrt[-4 a + b^2 - 4 a c])/( > 2 a)) + (-b + Sqrt[4 a + b^2 - 4 a c])/( > 2 a))/(-((-b + Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + (-b + > Sqrt[4 a + b^2 - 4 a c])/(2 a)), > ArcSin[Sqrt[((Sqrt[-4 a + b^2 - 4 a c] - Sqrt[ > 4 a + b^2 - 4 a c]) (b + Sqrt[-4 a + b^2 - 4 a c] + > 2 a x))/((Sqrt[-4 a + b^2 - 4 a c] + Sqrt[ > 4 a + b^2 - 4 a c]) (-b + Sqrt[-4 a + b^2 - 4 a c] - > 2 a x))]], (Sqrt[-4 a + b^2 - 4 a c] + Sqrt[ > 4 a + b^2 - 4 a c])^2/(Sqrt[-4 a + b^2 - 4 a c] - Sqrt[ > 4 a + b^2 - 4 a c])^2])/(-((-b + > Sqrt[-4 a + b^2 - 4 a c])/(2 a)) + (-b + Sqrt[ > 4 a + b^2 - 4 a c])/(2 a)))))/(Sqrt[-1 + c^2 + > 2 b c x + (b^2 + 2 a c) x^2 + 2 a b x^3 + > a^2 x^4] (c + x (b + a x)) Sqrt[-1 + c^2 + b^2 x^2 + 2 a b x^3 + > a^2 x^4 + 2 c x (b + a x)]) > How's that? I then use another system to do this issue with the following codes: theta := arcsin(a*x^2+b*x+c); int(tan(theta),x); The result is also very long, but I don't know whether both are equivalent. Thanks a lot. -- Hongyi Zhao <hongyi.zhao at gmail.com> Xinjiang Technical Institute of Physics and Chemistry Chinese Academy of Sciences GnuPG DSA: 0xD108493 2009-5-6
- References:
- Do some definite integral calculation.
- From: Hongyi <hongyi.zhao@gmail.com>
- Do some definite integral calculation.