A Complete Guide to Improper Integrals

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0:00The integral of $1/(x-1)$³ from 0 to infinity has both an infinite endpoint and an interior singularity at $x=1$. Treating only the infinite endpoint misses an essential condition.

4:30Split at $x=1$ and use separate one-sided limits for the left and right pieces, as well as a limit at infinity. An ordinary improper integral converges only if every required piece converges finitely.

7:30The antiderivative is −1/[$2(x-1)$²]. Approaching the singularity from the left makes the left integral tend to −infinity; the right integral tends to +infinity. They cannot be canceled to assign an ordinary integral value.

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