Can't integrate sqrt(a+b*cos(t)+c*cos(2t))
- To: mathgroup at smc.vnet.net
- Subject: [mg90775] Can't integrate sqrt(a+b*cos(t)+c*cos(2t))
- From: Valeri Astanoff <astanoff at gmail.com>
- Date: Wed, 23 Jul 2008 06:27:06 -0400 (EDT)
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