5. Improper Integral Tester
Improper integrals are defined using limits. This page summarizes Type I/II, convergence tests, and absolute vs conditional convergence.
Type I (infinite bounds)
If a bound is infinite, the integral is defined as a limit of proper integrals:
\[
\int_a^{\infty} f(x)\,dx \;=\; \lim_{B\to\infty}\int_a^{B} f(x)\,dx
\]
\[
\int_{-\infty}^{b} f(x)\,dx \;=\; \lim_{A\to-\infty}\int_A^{b} f(x)\,dx
\]
The improper integral converges if the limit exists and is finite; otherwise it diverges.
Type II (discontinuity / singularity)
If \(f(x)\) is not finite at some point \(c\) in \([a,b]\) (or at an endpoint), we split and take limits:
\[
\int_a^b f(x)\,dx
\;=\;
\lim_{\varepsilon\to 0^+}
\left(
\int_a^{c-\varepsilon} f(x)\,dx
+
\int_{c+\varepsilon}^{b} f(x)\,dx
\right)
\]
If there are multiple singular points, split at each one. The integral converges only if every resulting improper piece converges.
p-integral tests (quick recognition)
These are the most common “instant” convergence checks:
\[
\int_1^\infty \frac{1}{x^p}\,dx
\text{ converges iff } p>1
\]
\[
\int_0^1 \frac{1}{x^p}\,dx
\text{ converges iff } p<1
\]
\[
\int \frac{1}{|x-c|^p}\,dx \text{ near } x=c
\text{ converges iff } p<1
\]
The calculator may show a heuristic \(p\)-estimate based on numeric sampling if a pattern is not obvious.
Comparison and limit comparison
If \(0 \le f(x) \le g(x)\) for large \(x\) (or near a singularity), then:
\[
\int g \text{ converges } \Rightarrow \int f \text{ converges}
\qquad
\int f \text{ diverges } \Rightarrow \int g \text{ diverges}
\]
For positive functions, the limit comparison test says:
\[
\lim_{x\to\infty}\frac{f(x)}{g(x)} = L,\; 0
Absolute vs conditional convergence
The improper integral \(\int f(x)\,dx\) is:
- Absolutely convergent if \(\int |f(x)|\,dx\) converges.
- Conditionally convergent if \(\int f(x)\,dx\) converges but \(\int |f(x)|\,dx\) diverges.
Oscillatory integrals (e.g. \(\int_1^\infty \sin(x)/x\,dx\)) are classic conditional-convergence cases.
Convergence graph (partial integrals)
The calculator plots the sequence used in the definition:
- Type I: partial sums \(I(B)=\int_a^B f(x)\,dx\) as \(B\to\infty\) (or \(A\to-\infty\)).
- Type II: \(\varepsilon\)-partials as \(\varepsilon\to 0^+\) (shown against \(-\log_{10}\varepsilon\)).
Stabilization of this graph indicates convergence; persistent drift or blow-up indicates divergence.
