       Re: can not understand the symbol or the equation for an integration result

• To: mathgroup at smc.vnet.net
• Subject: [mg122237] Re: can not understand the symbol or the equation for an integration result
• From: Patrick Scheibe <pscheibe at trm.uni-leipzig.de>
• Date: Sat, 22 Oct 2011 06:05:27 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <201110211024.GAA17725@smc.vnet.net>

```Hi,

if you try a FullSimplify on your integration-result, you'll see that
those imaginary parts chancel out.

Cheers
Patrick

On Oct 21, 2011, at 12:24 PM, Jing wrote:

> Hi,
>
> I am trying to integrate a equation,
> 1/72 (8 Sqrt r^3 -
>   Sqrt r^2 Sqrt[r^2 + 2 y (y - Sqrt Sqrt[r^2 - y^2])] -
>   2 Sqrt y^2 Sqrt[r^2 + 2 y (y - Sqrt Sqrt[r^2 - y^2])] +
>   6 y Sqrt[(r^2 - y^2) (r^2 + 2 y (y - Sqrt Sqrt[r^2 - y^2]))])
>
> But the result is kind of confusing.First, the variable y is changed to another variable y -> -r Sin[t].
> exp1 = Simplify[1/72 (8 Sqrt r^3 -
>   Sqrt r^2 Sqrt[r^2 + 2 y (y - Sqrt Sqrt[r^2 - y^2])] -
>   2 Sqrt y^2 Sqrt[r^2 + 2 y (y - Sqrt Sqrt[r^2 - y^2])] +
>   6 y Sqrt[(r^2 - y^2) (r^2 + 2 y (y - Sqrt Sqrt[r^2 - y^2]))])
> /. {y -> -r Sin[t]}, r > 0 && 0 < t < Pi/6]
>
> result:-(1/72) r^3 (3 Sin[2 t] Sqrt[2 - Cos[2 t] + Sqrt Sin[2 t]] +
>   Sqrt (-8 + Sqrt[1 + 2 Sin[t]^2 + Sqrt Sin[2 t]] +
>      2 Sin[t]^2 Sqrt[1 + 2 Sin[t]^2 + Sqrt Sin[2 t]]))
>
> Then,do integration :
> m19 = Integrate[-r Cos[t] exp1, t, Assumptions -> {r > 0}]
> final result:
> 1/72 r^4 (-8 Sqrt Sin[t] + (3 t (Sqrt Cos[t] + 3 Sin[t]))/(
>   2 Sqrt[2 - Cos[2 t] + Sqrt Sin[2 t]]) - (
>   Sqrt[2 - Cos[2 t] +
>     Sqrt Sin[2 t]] (6 (I + Sqrt) Cos[t] +
>      3 (I + Sqrt) Cos[
>        3 t] + (3 + I Sqrt) (-8 Sin[t] + Sin[3 t])))/(
>   8 (I + Sqrt)))
>
> I don't know what the I is. Is it the imaginary number? Why it can appear?
>
> Thanks
>

```

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