Improper Integrals Math Example 1

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

Example 1

easy
Evaluate ∫1∞1x2 dx\displaystyle\int_1^{\infty} \frac{1}{x^2}\,dx.

Solution

  1. 1
    Replace the infinite upper limit with a variable: ∫1∞1x2 dx=lim⁡b→∞∫1bx−2 dx\int_1^{\infty}\frac{1}{x^2}\,dx = \lim_{b\to\infty}\int_1^b x^{-2}\,dx
  2. 2
    Integrate x−2x^{-2}: =lim⁡b→∞[−1x]1b=lim⁡b→∞(−1b+1)= \lim_{b\to\infty}\left[-\frac{1}{x}\right]_1^b = \lim_{b\to\infty}\left(-\frac{1}{b} + 1\right)
  3. 3
    Take the limit as b→∞b \to \infty: since 1b→0\frac{1}{b} \to 0, the integral converges to 11.

Answer

11
Replace ∞\infty with bb, integrate, take the limit. The 1/b1/b term vanishes.

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 2 hard

Evaluate [formula] (Type II).

Example 3 easy

Does [formula] converge or diverge?

Example 4 medium

Evaluate [formula].

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