# How do you test the improper integral #int x^-2dx# from #[-1,1]# and evaluate if possible?

The improper integral:

is divergent.

so that:

Now pose:

So that:

then the improper integral:

is divergent.

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To test the improper integral ∫x^(-2)dx from -1 to 1, first, identify if there are any singularities within the interval of integration. In this case, x^(-2) becomes undefined at x = 0, which is within the interval [-1, 1]. Then, split the integral into two parts: ∫x^(-2)dx from -1 to 0 and ∫x^(-2)dx from 0 to 1. Evaluate each part separately. The integral of x^(-2)dx is -x^(-1), so integrating from -1 to 0 yields -(-1)^(-1) - (-0)^(-1) = -(-1) - (-∞) = 1 + ∞ = ∞. Integrating from 0 to 1 yields (-1)^(-1) - (0)^(-1) = -1 - ∞ = -∞. Since both parts diverge, the original improper integral diverges.

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