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Re: Bug of Integrate

Miguel <misvrne at> wrote:
> When I try to calculate the integral
> Integrate[Sqrt[1/Cos[t]^2]*3*Cos[t],{t,0,2Pi}]  Mathematica 6.0.1
> yields -6*Pi.

If so, then that is clearly a bug. But in version 5.2, the result is

In[1]:= Integrate[Sqrt[1/Cos[t]^2]*3*Cos[t],{t,0,2Pi}]

Out[1]= 0

> Simplifying the expresion resultrs Integrate[3,{t,0,2*Pi}] and It is
> clear that the correct solution is 6*Pi.

No. Simplifying the integrand does not give 3, rather

In[2]:= Simplify[Sqrt[1/Cos[t]^2]*3*Cos[t], Element[t,Reals]]

Out[2]= 3 Abs[Sec[t]] Cos[t]

which BTW is the same as 3 Sign[Cos[t]] for real t except when Cos[t]==0.

However, related to the above, version 5.2 does give an incorrect result
for a definite integral with a symbolic real limit. Whether this error
still exists in version 6, I don't know:

In[3]:= Assuming[Element[x,Reals],Integrate[3*Sign[Cos[t]],{t,0,x}]]

Out[3]= 3 If[x > 0, x Abs[Cos[x]] Sec[x],
Integrate[Sign[Cos[t]], {t, 0, x}, Assumptions -> x <= 0]]

The above is incorrect for x > Pi/2. A correct result would have been

3 Sign[Cos[x]] (x - Pi Floor[x/Pi + 1/2])

for all real x.

David W. Cantrell

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