Improper integrals

Japanese school year: University year 1

What you learn

You will learn how to evaluate improper integrals where the interval of integration extends to infinity or the integrand has singularities, using limit processes. These integrals are extensively used in computing expectations of continuous random variables and physical potentials. Proficiency in definite integrals and limits is required.

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Key points

This improper integral calculates the area under curve $f(x)$ over an interval extending to infinity. First integrate over a finite interval $[a, b]$ , then take the limit as $b \to \infty$ .

\int_a^\infty f(x)\,dx = \lim_{b \to \infty} \int_a^b f(x)\,dx

Use this improper integral when the function $f(x)$ shoots to infinity at the left endpoint $x = a$ . Integrate starting slightly to the right at $a + \varepsilon$ , then take the limit as $\varepsilon \to +0$ .

\int_a^b f(x)\,dx = \lim_{\varepsilon \to +0} \int_{a+\varepsilon}^b f(x)\,dx

This formula computes the improper integral of $1/x^p$ from 1 to infinity. The area converges to $1/(p-1)$ only when power $p > 1$ ; otherwise, it diverges to infinity.

\int_1^\infty \frac{1}{x^p}\,dx = \frac{1}{p-1} \quad (p > 1)

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Practice 1Coming soon