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Re: a suprising result from Integrate (Null appeared in the
*To*: mathgroup at smc.vnet.net
*Subject*: [mg74505] Re: a suprising result from Integrate (Null appeared in the
*From*: Bhuvanesh <lalu_bhatt at yahoo.com>
*Date*: Fri, 23 Mar 2007 19:11:00 -0500 (EST)
Yes, this was reported and fixed quite a while back, and should be fine in the next version. Another example of the bad behavior was Integrate[Sqrt[Sin[x] + Cos[x]], x]. For your indefinite integrals I currently get, in the development build:
In[1]:= Integrate[(1 - Sin[x])^(1/4), x] //InputForm
Out[1]//InputForm=
(4*EllipticE[(Pi + 2*x)/8, 2]*(2 - 2*Sin[x])^(1/4))/Sqrt[Cos[x/2] - Sin[x/2]]
In[2]:= D[%, x] //Simplify //InputForm
Out[2]//InputForm= (1 - Sin[x])^(1/4)
In[3]:= Integrate[(1 - Sin[x])^(1/5), x] //InputForm
Out[3]//InputForm=
((-1/4 + I/4)*(-20*(-1)^(1/4)*2^(1/10)*E^(I*x)*(1 - I/E^(I*x))^(3/5)*
Hypergeometric2F1[-1/5, 3/5, 4/5, I/E^(I*x)]*Sin[(Pi - 2*x)/4]^(3/5) +
5*(-1)^(3/4)*2^(1/10)*(1 - I/E^(I*x))^(3/5)*Hypergeometric2F1[3/5, 4/5,
9/5, I/E^(I*x)]*Sin[(Pi - 2*x)/4]^(3/5) +
(2 + 2*I)*(((-1)^(1/4)*(-I + E^(I*x)))/E^((I/2)*x))^(3/5)*
(-5*I + 5*E^(I*x) - (2 - 2*I)*Sqrt[2]*E^((I/2)*x)*Cos[(Pi - 2*x)/4]*
Hypergeometric2F1[1/2, 3/10, 3/2, Cos[(Pi - 2*x)/4]^2]*
(Sin[(Pi - 2*x)/4]^2)^(3/10)))*(1 - Sin[x])^(1/5))/
((-I + E^(I*x))*(((-1)^(1/4)*(-I + E^(I*x)))/E^((I/2)*x))^(3/5))
In[4]:= D[%, x] //FullSimplify //InputForm
Out[4]//InputForm= (1 - Sin[x])^(1/5)
Thanks for the report, and I'm sorry for any inconvenience due to this issue.
Bhuvanesh,
Wolfram Research
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