Problem 34 - Calculus for Biology and Medicine

Question

Determine the constant \(c\) so that $$ \int_{-\infty}^{\infty} \frac{c}{1+x^{2}} d x=1 $$

Step-by-Step Solution

Answer

The constant \(c\) is \(\frac{1}{\pi}\).

Step 1: Recognize the Integral Type

The given integral is from \(-\infty\) to \(\infty\) of a function of the form \(\frac{1}{1+x^2}\). This function is known as the standard Cauchy distribution, which has an integral equal to \(\pi\) over the entire real line.

Step 2: Substitute the Given Function

The integral is given as \(\int_{-\infty}^{\infty} \frac{c}{1+x^2} \, dx = 1\). Here, \(\frac{c}{1+x^2}\) is a scaled version of the standard Cauchy distribution \(\frac{1}{1+x^2}\).

Step 3: Use the Integral of Standard Cauchy Distribution

For the standard Cauchy distribution, \(\int_{-\infty}^{\infty} \frac{1}{1+x^2} \, dx = \pi\). Hence, \(c\) times this integral must equal 1: \(c \cdot \pi = 1\).

Step 4: Solve for the Constant c

From the equation \(c \cdot \pi = 1\), solve for \(c\) by dividing both sides by \(\pi\), giving \(c = \frac{1}{\pi}\).

Key Concepts

Integral Calculus

Integral calculus is a huge part of mathematics, especially when dealing with areas, volumes, and other concepts that involve sum aggregation. One of the most important goals of integral calculus is to compute the concept of an integral, which represents accumulation or total sum.

Integrals can be thought of as the area under a curve on a graph. When a function is defined over an interval, the definite integral computes the accumulated value across that interval. There are two types of integrals commonly encountered:

• Definite Integrals: Represented as \(\int_a^b f(x)\,dx\), this computes the accumulated value from point \(a\) to \(b\).
• Indefinite Integrals: Represented as \(\int f(x)\,dx\), this finds a general form for the accumulation without specific bounds.

In the discussed problem, we handle the integral from \(-\infty\) to \(\infty\), known as an improper integral because it involves infinite bounds. This kind of improper integral often appears in probability and statistical distributions, especially in continuous cases.

Cauchy Distribution

The Cauchy distribution is a fundamental concept in statistics and probability theory. It's a type of probability distribution known for its heavy tails and undefined mean and variance. The standard form of the Cauchy distribution is expressed through the function \(\frac{1}{1+x^2}\).

This distribution is particularly useful because:

• Its integral across the entire real line \((-\infty\) to \(\infty\)) is equal to \(\pi\), which makes it straightforward to use when dealing with normalization.
• It does not fall into the same category as the normal distribution because its statistical qualities are not defined.
• The Cauchy distribution is linked to the inverse tangent function, which often surfaces in calculus problems.

The problem involving the Cauchy distribution required identifying the need to normalize such a distribution to find a constant \(c\), resulting in an integral value of one over its support; this entails moderation by \(\pi\), yielding \(c = \frac{1}{\pi}\).

Improper Integrals

Improper integrals are those integrals where either the interval is infinite, or the function has infinite discontinuities. These integrals often arise because we want to know the total accumulation of values under a curve that extends beyond the limits of standard, finite intervals.

In the given exercise, the integral is improper because it runs from \(-\infty\) to \(\infty\). To handle these:

• We typically use limits to define what happens at the boundaries: \(\lim_{b \to \infty} \int_{-b}^{b} \frac{c}{1+x^{2}}\, dx\).
• If these limits exist and are finite, the integral is convergent.
• If they diverge, meaning they don't settle into a fixed number, the integral is divergent.

Understanding improper integrals is critical in calculus as they help to evaluate the behavior of physical systems over unbounded domains. The fact that the integral from \(-\infty\) to \(\infty\) for the scaled Cauchy distribution could be made to equal \(1\) by choosing \(c=\frac{1}{\pi}\) indicates convergence, and thus was crucial to solving the problem.