O que seria uma integral impropria?

An improper integral generally looks like this (this is called improper Riemann integral of the first kind): $$\lim_{a \to \infty} \int_b^a f(x)$$ or this (Improper Riemann integral of the second kind): $$\lim_{a \to c} \int_b^a f(x)$$ where $c$ is a real number.

But the part after $\lim$ looks just like a proper integral!

You may want to ask, why we have to include a $\lim$ in front of the integral?

Recall how we obtain a Riemann sum. We divide the interval in small segments, and approximate the value by assuming each interval correspond to a rectangle.

Source: Wikipedia.org

When the interval we integrate is infinitely long, or if the function approach infinity at some point, the area of that particular rectangle will become infinitely large. Therefore, we cannot actually take the integral at that point – but we can get closer and closer to that endpoint and see if we are approaching a finite area.

Improper Integral of the First Kind

For improper integral of the first kind, dividing the interval into a large number of segments doesn’t ensure that each segment is small. Since one or both of the endpoints of the interval are at infinity, the interval is infinitely long. We can’t really divide an infinitely long interval into finitely many tiny segments.

However, by taking the limit of the endpoint, and when the area is integrable, the limit of the integral will converge to the actual area under the function.

For example, $$\int_1^{\infty} \frac{1}{x^2}$$ would be an improper integral of the first kind, and we have to evaluate it as $$\lim_{a \to \infty} \int_1^{a} \frac{1}{x^2}$$

Improper Integral of the Second Kind

Think about an improper integral of the second kind as an improper integral of the first kind, rotated 90 degrees.

If you try to take the Riemann sum, since the endpoint is mapped to infinity, either the last rectangle has an infinite area or the leftover region is unbounded. Therefore, we don’t place or endpoints at the discontinuities – but rather let the endpoints approach the discontinuities.