# How do you evaluate #int cot(x)# from 0 to 2pi?

The integral diverges.

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This improper integral does not converge.

To attempt to evaluate the integral, we need to find the two improper integrals

But this limit does not exist, so the integral diverges.

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To evaluate the integral of cot(x) from 0 to 2π, you can rewrite cot(x) as 1/tan(x), then integrate 1/tan(x) with respect to x from 0 to 2π. Since cot(x) is the reciprocal of tan(x), integrating 1/tan(x) is equivalent to integrating tan(x)^(-1). The integral of tan(x)^(-1) is ln|sec(x)|. So, integrating 1/tan(x) from 0 to 2π gives ln|sec(2π)| - ln|sec(0)|. Since sec(0) = 1 and sec(2π) = 1, the result is ln(1) - ln(1) = 0. Therefore, the integral of cot(x) from 0 to 2π is 0.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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