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:

> 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