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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


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