Improper Integrals Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

easy
Does ∫1∞1x dx\displaystyle\int_1^{\infty} \frac{1}{x}\,dx converge or diverge?

Solution

  1. 1
    lim⁡b→∞[ln⁡x]1b=lim⁡b→∞ln⁡b=∞\lim_{b\to\infty}[\ln x]_1^b = \lim_{b\to\infty}\ln b = \infty. Diverges.

Answer

Diverges.
ln⁡b→∞\ln b \to \infty, so no finite limit exists. This is the integral analog of the divergent harmonic series.

About Improper Integrals

Integrals where the interval of integration is infinite (Type I: ∫a∞f(x) dx\int_a^{\infty} f(x)\,dx) or the integrand has an infinite discontinuity on the interval (Type II: ∫abf(x) dx\int_a^b f(x)\,dx where ff blows up at some point in [a,b][a, b]). Evaluated as limits of proper integrals.

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More Improper Integrals Examples

Example 1 easy

Evaluate [formula].

Example 2 hard

Evaluate [formula] (Type II).

Example 4 medium

Evaluate [formula].

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