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

• To: mathgroup at smc.vnet.net
• Subject: [mg90801] Re: Can't integrate sqrt(a+b*cos(t)+c*cos(2t))
• From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
• Date: Thu, 24 Jul 2008 05:04:57 -0400 (EDT)
• References: <g6710s\$sb6\$1@smc.vnet.net>

```Hi,

expr = 5 - 4*Cos[t] + Cos[2*t] // TrigExpand
Integrate[Sqrt[expr], {t, 0, Pi}]

??

Regards
Jens

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
>

```

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