Re: Do some definite integral calculation.
- To: mathgroup at smc.vnet.net
- Subject: [mg99488] Re: [mg99393] Do some definite integral calculation.
- From: DrMajorBob <btreat1 at austin.rr.com>
- Date: Wed, 6 May 2009 05:28:59 -0400 (EDT)
- References: <200905050938.FAA20515@smc.vnet.net>
- Reply-to: drmajorbob at bigfoot.com
> theta := arcsin(a*x^2+b*x+c);
> int(tan(theta),x);
I probably should have done it that way in Mathematica:
theta = ArcSin[a*x^2 + b*x + c];
Integrate[Tan[theta], x]
or just
Integrate[Tan@ArcSin[a*x^2 + b*x + c], x]
There are still multiple possible answers, and I'm not sure which method
gives a simpler one.
Answers can differ by a constant, and in general, Integrate outputs can
also be discontinuous on the real line, so that the Fundamental Theorem of
Integral Calculus doesn't apply.
Bobby
On Tue, 05 May 2009 21:24:47 -0500, <hongyi.zhao at gmail.com> wrote:
> 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 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.
--
DrMajorBob at bigfoot.com
- References:
- Do some definite integral calculation.
- From: Hongyi <hongyi.zhao@gmail.com>
- Do some definite integral calculation.