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>

```> 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:
>
>> 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

```

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