Re: Do some definite integral calculation.
- To: mathgroup at smc.vnet.net
- Subject: [mg99464] Re: [mg99393] Do some definite integral calculation.
- From: DrMajorBob <btreat1 at austin.rr.com>
- Date: Wed, 6 May 2009 05:24:29 -0400 (EDT)
- References: <200905050938.FAA20515@smc.vnet.net>
- Reply-to: drmajorbob at bigfoot.com
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)] (First could be used, rather than Last.) 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? Bobby On Tue, 05 May 2009 04:38:11 -0500, Hongyi <hongyi.zhao at gmail.com> wrote: > Hi all, > > I've the following equations: > > > \[\sin \left( \theta \right) = a{x^{^2}} + bx + c\] > > and > > \[y = \int_0^x {\tan } \left( \theta \right)dx\] > > I want to obtain the expression of y as the function of x. How should > I write the code within mathematica? > > Thanks in advance. > -- DrMajorBob at bigfoot.com
- References:
- Do some definite integral calculation.
- From: Hongyi <hongyi.zhao@gmail.com>
- Do some definite integral calculation.