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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[3] r^3 - 
   Sqrt[3] r^2 Sqrt[r^2 + 2 y (y - Sqrt[3] Sqrt[r^2 - y^2])] - 
   2 Sqrt[3] y^2 Sqrt[r^2 + 2 y (y - Sqrt[3] Sqrt[r^2 - y^2])] + 
   6 y Sqrt[(r^2 - y^2) (r^2 + 2 y (y - Sqrt[3] 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[3] r^3 - 
   Sqrt[3] r^2 Sqrt[r^2 + 2 y (y - Sqrt[3] Sqrt[r^2 - y^2])] - 
   2 Sqrt[3] y^2 Sqrt[r^2 + 2 y (y - Sqrt[3] Sqrt[r^2 - y^2])] + 
   6 y Sqrt[(r^2 - y^2) (r^2 + 2 y (y - Sqrt[3] 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[3] Sin[2 t]] + 
   Sqrt[3] (-8 + Sqrt[1 + 2 Sin[t]^2 + Sqrt[3] Sin[2 t]] + 
      2 Sin[t]^2 Sqrt[1 + 2 Sin[t]^2 + Sqrt[3] Sin[2 t]]))

Then,do integration :
m19 = Integrate[-r Cos[t] exp1, t, Assumptions -> {r > 0}]
final result: 
1/72 r^4 (-8 Sqrt[3] Sin[t] + (3 t (Sqrt[3] Cos[t] + 3 Sin[t]))/(
   2 Sqrt[2 - Cos[2 t] + Sqrt[3] Sin[2 t]]) - (
   Sqrt[2 - Cos[2 t] + 
     Sqrt[3] Sin[2 t]] (6 (I + Sqrt[3]) Cos[t] + 
      3 (I + Sqrt[3]) Cos[
        3 t] + (3 + I Sqrt[3]) (-8 Sin[t] + Sin[3 t])))/(
   8 (I + Sqrt[3])))

I don't know what the I is. Is it the imaginary number? Why it can appear? 

Thanks



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