e*******n 发帖数: 174 | 1 No Chinese input here. Sorry about it.
I've got an integral that I was able to integrate with Mathematica. It does
not appear to be difficult. So I tried to solve it manually. I am not able
to figure it out. For the sake of my curiosity, I wonder if anybody could
help me out. Any hints would be greatly appreciated.
The integral has the following form:
Integrate[Cos[x]^2*Log[-a*Cos[x] +Sqrt[1-a^2*Sin[x]^2]], {x, 0, 2*Pi}]
where "a" is a real positive number less than 1.
Again, thanks for any hint | s******1 发帖数: 969 | 2 积分分成[0,Pi],[Pi,2Pi],
Integrate[Cos[x]^2*Log[-a*Cos[x] +Sqrt[1-a^2*Sin[x]^2]], {x, Pi, 2*Pi}]
= Integrate[Cos[x]^2*Log[a*Cos[x] +Sqrt[1-a^2*Sin[x]^2]], {x,0, Pi}]
然后:
Integrate[Cos[x]^2*Log[-a*Cos[x] +Sqrt[1-a^2*Sin[x]^2]], {x, Pi, 2*Pi}]
=Integrate[Cos[x]^2*Log[1-a^2], {x,0, Pi}] | e*******n 发帖数: 174 | 3
~~~~~~~~~~~~~~~~~~~~~~~~~~?
How did you do this? I am lost. :)
【在 s******1 的大作中提到】![](/moin_static193/solenoid/img/up.png) : 积分分成[0,Pi],[Pi,2Pi], : Integrate[Cos[x]^2*Log[-a*Cos[x] +Sqrt[1-a^2*Sin[x]^2]], {x, Pi, 2*Pi}] : = Integrate[Cos[x]^2*Log[a*Cos[x] +Sqrt[1-a^2*Sin[x]^2]], {x,0, Pi}] : 然后: : Integrate[Cos[x]^2*Log[-a*Cos[x] +Sqrt[1-a^2*Sin[x]^2]], {x, Pi, 2*Pi}] : =Integrate[Cos[x]^2*Log[1-a^2], {x,0, Pi}]
| s******1 发帖数: 969 | 4 A==Integrate[Cos[x]^2*Log[-a*Cos[x] +Sqrt[1-a^2*Sin[x]^2]], {x, Pi, 2*Pi}]:
Introduce y=x-Pi so the range is [0,Pi].
Cos[x]=Cos[y+Pi]=-Cos[y]. Sin[x]=Sin[x+Pi]=-Sin[x].
A=Integrate[Cos[y]^2*Log[a*Cos[y] +Sqrt[1-a^2*Sin[y]^2]], {y,0, Pi}]
=Integrate[Cos[x]^2*Log[a*Cos[x] +Sqrt[1-a^2*Sin[x]^2]], {x,0, Pi}];
B==Integrate[Cos[x]^2*Log[-a*Cos[x] +Sqrt[1-a^2*Sin[x]^2]], {x, 0, Pi}];
Integrate[Cos[x]^2*Log[-a*Cos[x] +Sqrt[1-a^2*Sin[x]^2]], {x, Pi, 2*Pi}]
= A+B
=Integrate[Cos[x]^2*{Log[a*Cos[x] +Sqrt[1- | n*n 发帖数: 202 | 5 晕了....
[1
【在 s******1 的大作中提到】![](/moin_static193/solenoid/img/up.png) : A==Integrate[Cos[x]^2*Log[-a*Cos[x] +Sqrt[1-a^2*Sin[x]^2]], {x, Pi, 2*Pi}]: : Introduce y=x-Pi so the range is [0,Pi]. : Cos[x]=Cos[y+Pi]=-Cos[y]. Sin[x]=Sin[x+Pi]=-Sin[x]. : A=Integrate[Cos[y]^2*Log[a*Cos[y] +Sqrt[1-a^2*Sin[y]^2]], {y,0, Pi}] : =Integrate[Cos[x]^2*Log[a*Cos[x] +Sqrt[1-a^2*Sin[x]^2]], {x,0, Pi}]; : B==Integrate[Cos[x]^2*Log[-a*Cos[x] +Sqrt[1-a^2*Sin[x]^2]], {x, 0, Pi}]; : Integrate[Cos[x]^2*Log[-a*Cos[x] +Sqrt[1-a^2*Sin[x]^2]], {x, Pi, 2*Pi}] : = A+B : =Integrate[Cos[x]^2*{Log[a*Cos[x] +Sqrt[1-
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