Improper Integrals with Infinite Intervals

Definitions, Step-by-Step Worked Examples, Practice Exercises, and Solutions

Define and calculate improper integrals with infinite limits of integration through examples with detailed solutions and explanations. More exercises with solutions are included.

Definition of Improper Integrals with Infinite Intervals [1] [2]

We consider 3 types of integrals with infinite interval(s):

1 - The Upper Limit of the Integral is Infinite

An integral of the form \( \displaystyle \int_a^{\infty} f(x) \, dx \) exists and is convergent if \( \int_a^{b} f(x) \, dx \) exists for all \( b \ge a \) and the limit:

\[ \lim_{b \to \infty} \int_a^{b} f(x) \, dx \]

exists and is finite. We then write:

\[ \int_a^{\infty} f(x) \, dx = \lim_{b \to \infty} \int_a^{b} f(x) \, dx \]

2 - The Lower Limit of the Integral is Infinite

An integral of the form \( \displaystyle \int_{-\infty}^a f(x) \, dx \) exists and is convergent if \( \int_b^{a} f(x) \, dx \) exists for all \( b \le a \) and the limit:

\[ \lim_{b \to -\infty} \int_b^{a} f(x) \, dx \]

exists and is finite. We then write:

\[ \int_{-\infty}^a f(x) \, dx = \lim_{b \to -\infty} \int_b^{a} f(x) \, dx \]

3 - Both Limits of the Integral are Infinite

An integral of the form \( \displaystyle \int_{-\infty}^{\infty} f(x) \, dx \) exists and is convergent if both integrals:

\[ \int_{-\infty}^a f(x) \, dx \quad \text{and} \quad \int_a^{\infty} f(x) \, dx \]

are convergent for any real number \( a \). We then write:

\[ \int_{-\infty}^{\infty} f(x) \, dx = \int_{-\infty}^a f(x) \, dx + \int_a^{\infty} f(x) \, dx \]

Note: The interval of integration is split at \( x = a \), resulting in the sum of two integrals, each having only one infinite limit.

unit step function
Figure 1. Unit Step Function Reference.

Examples and Their Solutions

Example 1

Evaluate the integral given below if possible:

\[ \int_1^{\infty} x \, dx \]

Example 2

Is the integral below convergent or divergent? Evaluate it if it is convergent.

\[ \int_1^{\infty} \dfrac{1}{x^2} \, dx \]

Example 3

Evaluate the integral given below if possible:

\[ \int_{-\infty}^0 \dfrac{1}{\sqrt{2 - 3x}} \, dx \]

Example 4

Evaluate the integral given below if possible:

\[ \int_{-\infty}^{\infty} x^5 e^{-x^6} \, dx \]

Example 5

For what values of \( p \) is the integral:

\[ \int_{e}^{\infty} \dfrac{1}{x(\ln x)^{p+1}} \, dx \]

convergent? Find its value in terms of \( p \).

Exercises

Evaluate each of the following integrals if possible:

  1. \( \displaystyle \int_1^{\infty} \dfrac{1}{(4x+1)^2} \, dx \)
  2. \( \displaystyle \int_{-\infty}^{\infty} \dfrac{x}{1+x^2} \, dx \)
  3. \( \displaystyle \int_{-\infty}^{-1} e^{2x} \, dx \)
  4. \( \displaystyle \int_{-\infty}^{\infty} \dfrac{7x^2}{3+x^6} \, dx \)
  5. \( \displaystyle \int_0^{\infty} \dfrac{e^{2x}}{e^{4x}+3} \, dx \)

More References and Links

  1. University Calculus - Early Transcendentals - Joel Hass, Maurice D. Weir, George B. Thomas, Jr., Christopher Heil - ISBN-13: 978-0134995540
  2. Calculus - Early Transcendentals - James Stewart - ISBN-13: 978-0-495-01166-8
  3. Integrals and their applications in calculus