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Section Improper Integrals

In our definition of the definite integral \(\dsp \int_a ^b f(x) \; dx\text{,}\) it was assumed that \(f\) was defined on all of \([a, b]\) where \(a\) and \(b\) were numbers. We wish to extend our definition to consider the possibilities where \(a\) and \(b\) are \(\pm \infty\) and where \(f\) might not be defined at every point in \([a,b].\) When any of these possibilities occur, the integral is called an improper integral.

Problem 5.40.

Let \(\dsp f(x) = \frac{1}{x^2}.\)

  1. Sketch a graph of \(f.\)

  2. Compute \(\dsp \int_1^{10} f(x) \; dx, \int_1^{50} f(x) \; dx,\) and \(\dsp \int_1^{100} f(x) \; dx.\)

  3. Compute \(\dsp \lim_{N \to \infty} \int_1^{N} f(x) \ dx.\)

Definition 5.41.

If \(a\) is a number and \(f\) is continuous for all \(x \ge a\text{,}\) then we define the improper integral \(\dsp \int_a^{\infty} f(x) \; dx\) to be \(lim_{N \to \infty} \int_a^{N} f(x) \ dx.\) If this limit exists, we say the integral converges and if it does not exists, we say the integral diverges.

Problem 5.42.

Write down definitions for \(\dsp \int_{-\infty}^a f(x) \; dx\) and \(\dsp \int_{-\infty}^{\infty} f(x) \; dx.\)

Problem 5.43.

Let \(\dsp f(x) = \frac{4}{(x-1)^2}.\)

  1. Sketch the graph of \(f.\)

  2. Evaluate \(\dsp \int_2^{100} f(x) \; dx\)

  3. Evaluate \(\dsp \int_2^{100,000} f(x) \; dx\)

  4. Evaluate \(\dsp \int_2^{\infty} f(x) \; dx\)

  5. Evaluate \(\dsp \int_a^{\infty} f(x) \; dx\) where \(a > 1\text{.}\)

Problem 5.44.

Sketch \(\dsp f(x) = \frac{1}{x-1}\) and evaluate \(\dsp \int_2^{\infty} f(x) \; dx\text{.}\)

Problem 5.45.

Sketch \(\dsp f(x) = \frac{1}{4+x^2}\) and evaluate \(\dsp \int_{-\infty}^{\infty} f(x) \; dx.\)

Problem 5.46.

Evaluate \(\dsp \int_{-\infty}^{-2} \frac{5}{x \sqrt{x^2-1}} \; dx\text{.}\)

Definition 5.47.

If \(f\) is continuous on the interval \([a, b)\text{,}\) \(\displaystyle{\lim_{x \to b^-} |f(x)| = \infty}\text{,}\) then we define \(\dsp \int_a^{b} f(x) \ dx = {\lim_{c \to b^-} \int_a ^c f(x) \; dx}\text{.}\)

Problem 5.48.

Definitions.

  1. Suppose \(f\) is continuous on the interval \((a, b]\) and \(\displaystyle{\lim_{x \to a^+} |f(x)| = \infty}\text{.}\) Write a definition for \(\dsp \int_a^{b} f(x) dx\text{.}\)

  2. Suppose \(f\) is continuous on the interval \([a, b]\) except at \(c\) where \(a \lt c \lt b.\) Suppose \(\dsp \lim_{x \to c-} = \dsp \lim_{x \to c+} f(x) = \pm \infty\text{.}\) Write a definition for \(\dsp \int_a^{b} f(x) dx\text{.}\)

Problem 5.49.

Evaluate each of the following improper integrals.

  1. \(\dsp \int_0^2 \frac{1}{(x-2)^2} \ dx\)

  2. \(\dsp \int_3^5 \frac{x}{4-x} \; dx\)

  3. \(\dsp \int_0^\frac{\pi}{2} \cot x \; dx\)

  4. \(\dsp \int_0^e \frac{\ln x}{x} \; dx\)

  5. \(\dsp \int_1^{\infty} \frac{\; dx}{-2x \sqrt{x^2-1}}\)

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