Improper integrals and limits of integration

Try

The integral of 1/x² from 1 to a finite R is 1−1/R. Letting R grow gives the improper integral 1. This is a limit of ordinary integrals on compact intervals. Infinity is not an endpoint at which a function is evaluated; the limiting notation abbreviates a separate convergence claim. Each finite-cutoff calculation must be valid before its limit is considered.

Predict

Predict the corresponding integral of 1/x. The finite value is log R, which grows without bound, so the improper integral diverges even though its integrand tends to zero. Compare the different decay rates. As with series, a vanishing integrand is not a sufficient tail-convergence test. Near a finite singular endpoint, the threshold changes because the interval lengths shrink rather than extend indefinitely.

Experiment

Organize the power examples.

Derive the table from the power antiderivative on each nonsingular finite interval, treating p=1 separately. Do not infer the second row by copying the first: the limiting variable approaches zero, reversing which exponents decay.

Understand

For a function integrable on every finite interval [a,R], define ∫ₐ∞f as lim as R→∞ of those cutoff integrals, if finite. At a singular left endpoint a, use lim as ε↓0 of ∫ from a+ε to b. If there are two infinite endpoints or an interior singularity, split at a finite regular point and require the relevant one-sided limits independently.

A comparison proof uses completeness. If 0≤f≤g and the improper integral of g is finite, the cutoff integrals of f increase and remain bounded above by that value. Their supremum is finite, and the monotone-limit argument proves convergence. Absolute improper integrability also implies signed convergence: the absolute integral bounds every finite signed tail, making cutoff values Cauchy. This is the integral analogue of the absolute-series argument.

Apply

Repair the cancellation ∫ from −R to R of x dx=0 as a proof of ordinary improper integrability on R. The positive tail diverges to positive infinity and the negative tail to negative infinity. Coupling their cutoffs gives a symmetric principal value, a different notion; independent one-sided limits do not exist. Likewise, evaluating −1/x across the singularity in 1/x² would hide two positive divergent contributions.

For multiple singularities, isolate each one before combining results. A finite sum of valid convergent pieces is allowed, while cancellation between divergent pieces is not an ordinary improper integral. State which limiting definition is intended rather than treating all formal expressions alike.

Classify an endpoint before using cancellation

Consider sin(t)/t^α on (0,1]. Although a sine appears, it does not oscillate in sign on this interval. Concavity of sine between zero and one gives sin(1)t ≤ sin(t), while sin(t) ≤ t follows by the mean value theorem. Dividing by the positive t^α places the integrand between positive multiples of t^(1−α). Thus both ordinary and absolute improper convergence hold exactly when α<2. At α=2 the logarithmic lower comparison diverges. For α<2 the omitted integral from zero to ε is at most ε^(2−α)/(2−α).

At infinity, exp(−t)/t^β is absolutely integrable for every real β. Choose an integer m≥max(0,−β). The exponential series gives exp(t/2) ≥ (t/2)^m/m!, so for t≥1,

The right side has tail integral 2^(m+1)m! exp(−R/2) beyond R≥1. This is a genuine bound for every fixed β; “exponentials dominate powers” is not a substitute for choosing the integer and checking its inequality.

Now replace sin(t) by sin(1/t). This changes the mechanism, not just the picture.

Oscillation gives a conditional range

Make the substitution u=1/t on each finite interval [ε,1] before taking a limit. It transforms the integral into the integral from 1 to 1/ε of sin(u)u^(α−2). Put p=2−α. If p>0, integration by parts on a finite tail [A,B] gives

The boundary term at B tends to zero, and the remaining cosine integral converges absolutely by power comparison. The resulting infinite signed tail has absolute value at most 2A^(−p). In the original variable, the omitted tail from zero to ε is therefore at most 2ε^(2−α). This proves ordinary convergence for α<2, including cases where the absolute integral fails.

For absolute convergence, α<1 suffices by |sin(1/t)|≤1 and the power test. To prove necessity, take disjoint positive blocks from 2πn+π/6 to 2πn+5π/6, where sin(u)≥1/2. Their weighted absolute integrals are bounded below by a positive constant times n^(α−2): the block length is 2π/3 and its endpoints are fixed multiples of n up to bounded factors. When α≥1 these lower bounds have divergent sum. Thus absolute convergence holds exactly for α<1.

When α≥2 the same positive blocks have integrals bounded below by π/3 (and growing if α>2), even though both block endpoints tend to infinity. This violates the Cauchy condition for an ordinary improper integral. It is not enough to inspect cutoff values only at multiples of a whole period, where cancellations can conceal the failure.

The complete classification for sin(1/t)/t^α is: absolute convergence for α<1; conditional convergence for 1≤α<2; divergence for α≥2. The plot and finitely many bounds illustrate this proof but cannot replace its quantifiers. The separate integral of exp(−t)/t^β has no conditional range because its integrand is positive.

Continue

Optional depth connects improper integrals to series through decreasing-function comparisons and to more advanced integration theories. This unit has developed Darboux integration and classical improper limits, not Lebesgue integration or a general interchange-of-limits theorem. Passing a limit through an integral needs hypotheses that will be studied in Analysis II.

Finish a proof portfolio with a completeness argument, a compactness application, a derivative theorem proof and an integral convergence classification. Explain where each uses a universal estimate rather than finite numerical evidence. The sequence establishes Real Analysis I practice; cumulative review and unfamiliar proof assessment remain separate from completing lesson pages.