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Re: Can't integrate sqrt(a+b*cos(t)+c*cos(2t))

  • To: mathgroup at smc.vnet.net
  • Subject: [mg90800] Re: Can't integrate sqrt(a+b*cos(t)+c*cos(2t))
  • From: Bob F <deepyogurt at gmail.com>
  • Date: Thu, 24 Jul 2008 05:04:46 -0400 (EDT)
  • References: <g6710s$sb6$1@smc.vnet.net>

On Jul 23, 4:26 am, Valeri Astanoff <astan... at gmail.com> wrote:
> Good day,
>
> Neither Mathematica 6 nor anyone here can integrate this:
>
> In[1]:= Integrate[Sqrt[5 - 4*Cos[t] + Cos[2*t]], {t, 0, Pi}]
> Out[1]= Integrate[Sqrt[5 - 4*Cos[t] + Cos[2*t]], {t, 0, Pi}]
>
> In[2]:= NIntegrate[Sqrt[5 - 4*Cos[t] + Cos[2*t]], {t, 0, Pi}]
> Out[2]= 6.72288
>
> I know the exact result:
>
> In[3]:=  (1/5^(3/4))*(Sqrt[2]*(10*EllipticE[(1/10)*(5 - Sqrt[5])] -
>         10*EllipticK[(1/10)*(5 - Sqrt[5])] + (5 + 3*Sqrt[5])*
>         EllipticPi[(1/10)*(5 - 3*Sqrt[5]), (1/10)*(5 - Sqrt[5])])=
)//N
> Out[3]= 6.72288
>
> but I would like to prove it.
>
> Thanks in advance to the samaritan experts...
>
> V.Astanoff

You can use the TrigExpand[] function for the Cos[2t] to get the
equivalent Cos[t]^2 - Sin[t]^2, and if you do this Mathematica 6.0.3
on a Mac comes up with

1/5 Sqrt[2 + 4 I] (-5 I EllipticE[-(3/5) - (4 I)/5] +
          (2 + I) Sqrt[5] EllipticE[-(3/5) + (4 I)/5] -
                 (12 - 4 I) EllipticK[-(3/5) - (4 I)/5] +
        (6 - 2 I) Sqrt[5] EllipticK[8/5 - (4 I)/5] +
               4 I Sqrt[5] EllipticPi[1/5 + (2 I)/5, -(3/5) + (4 I)/5]
+
                  (8 + 4 I) EllipticPi[1 - 2 I, -(3/5) - (4 I)/5])

The equivalence to your expression is left up to you...but this does
evaluate numerically to the same as what you had.

-Bob


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