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Consider the integral I=int(0)^(2pi)(dx)...

Consider the integral `I=int_(0)^(2pi)(dx)/(5-2cosx)`
Making the substitution `"tan"1/2x=t`, we have
`I=int_(0)^(2pi)(dx)/(5-2cosx)=int_(0)^(0)(2dt)/((1+t^(2))[5-2(1-t^(2))//(1+t^(2))])=0`
The result is obviously wrong, since the integrand is positive and consequently the integral of this function cannot be equal to zero. Find the mistake.

Text Solution

Verified by Experts

The correct Answer is:
2
Here the mistake lies in the substitution `"tan"1/2x=t`, because `"tan"1/2x` is discontinuous at `x=pi` which is a point in the interval `[0,2pi]`.
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