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.