Improper Integrals

The central idea

An integral is improper when an interval is unbounded or the integrand is unbounded. It is defined by a limit, such as $\int_a^\infty f=\lim_{b\to\infty}\int_a^bf$. An interior singularity requires splitting and separate one-sided limits. The integral converges only if every required limit is finite.

Definitions, hypotheses, and notation

The $p$-integral tests differ at zero and infinity: $\int_1^\infty x^{-p}dx$ converges for $p>1$, while $\int_0^1x^{-p}dx$ converges for $p<1$. The same formula has opposite thresholds because different endpoints cause the singular behavior.

A Cauchy principal value may assign a symmetric cancellation to some divergent expressions, but it is not the ordinary improper integral. Introductory convergence requires each one-sided piece separately finite, preventing positive and negative infinities from being combined.

Conceptual meaning

Improper notation abbreviates a limiting process; infinity is not a bound that can be substituted into an antiderivative. Convergence means finite signed accumulation, while divergence means at least one limiting contribution fails to settle.

A dependable method and decision rule

1. Locate every infinite endpoint and singular point.
2. Split at each interior singularity before computing.
3. Replace each improper piece by its own limit with a finite variable endpoint.
4. Evaluate the proper integral first, then take the limit.
5. Declare convergence only when all pieces have finite limits.

Fully worked example

Graphical or geometric meaning

An unbounded region can have finite area when its height decays sufficiently fast. Successive tails under $1/x^2$ shrink geometrically in scale, whereas logarithmic tail area under $1/x$ continues growing.

Common mistakes and why they fail

Verification and reasonableness checks

• Differentiate the antiderivative on each valid interval.
• Use comparison with a known $p$-integral.
• Confirm every one-sided limit at a singularity is handled separately.

Every improper feature gets its own limit

Replace an infinite endpoint by a finite variable and take a limit. An unbounded integrand at an interior point $c$ requires splitting at $c$ and evaluating left and right limits independently. Both must converge; cancellation across the singularity is not the ordinary improper integral. Split again for every additional singularity. The benchmarks $\int_1^\infty x^{-p}dx$ and $\int_0^1x^{-p}dx$ have opposite convergence conditions, so identify whether the issue is infinity or a finite endpoint. Write the limiting expression before using an antiderivative, and never substitute an infinity symbol as though it were a number. This setup makes the convergence claim logically visible.

For nonnegative integrands, comparison can decide convergence without finding an antiderivative. Choose a benchmark with the same problematic endpoint, establish the inequality only near that endpoint, and handle the remaining bounded interval as an ordinary proper integral.

Practice

1. Evaluate $\int_1^\infty x^{-3}dx$.
2. Does $\int_1^\infty x^{-1/2}dx$ converge?
3. Does $\int_0^1x^{-1/2}dx$ converge?

Connections and next steps

• Integral Test
• Comparison Test
• Infinite Series