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>

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?
>