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

• To: mathgroup at smc.vnet.net
• Subject: [mg122220] can not understand the symbol or the equation for an integration result
• From: Jing <jing.guo89 at yahoo.com>
• Date: Fri, 21 Oct 2011 06:24:05 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com

```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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