Mathematics

Mathematics

Problem

For ∫−∞7dx(x−3)2\displaystyle \int_{-\infty}^{7} \frac{dx}{(x-3)^{2}}, write the separate improper integrals that must be evaluated to test convergence before any limits are taken.
Intuitive explanation

This integral has a singularity at x=3x = 3, which lies inside the integration interval [−∞,7][-\infty, 7]. The key insight is that you can't take a single limit across a point where the integrand blows up—you must split the integral at that singularity and evaluate each piece independently with its own limit. Only if both pieces converge can you claim the whole integral converges.

Intuitive explanation

This integral has a singularity at x=3x = 3, which lies inside the integration interval [−∞,7][-\infty, 7]. The key insight is that you can't take a single limit across a point where the integrand blows up—you must split the integral at that singularity and evaluate each piece independently with its own limit. Only if both pieces converge can you claim the whole integral converges.

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