Re: Can't integrate sqrt(a+b*cos(t)+c*cos(2t))
- To: mathgroup at smc.vnet.net
- Subject: [mg90784] Re: Can't integrate sqrt(a+b*cos(t)+c*cos(2t))
- From: "Kevin J. McCann" <Kevin.McCann at umbc.edu>
- Date: Thu, 24 Jul 2008 04:52:22 -0400 (EDT)
- Organization: University System of Maryland
- References: <g6710s$sb6$1@smc.vnet.net>
I actually get:
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])
which evaluates to your numerical answer below,
Valeri Astanoff 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
>