Problem 30 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Determine whether the integral is convergent or divergent. Evaluate all convergent integrals. Be efficient. If \(\lim _{x \rightarrow \infty} \neq 0\), then \(\int_{a}^{\infty} f(x) d x\) is divergent. \(\int_{1}^{\infty} \frac{\arctan x}{1+x^{2}} d x\)

Step-by-Step Solution

Answer

The integral \(\int_{1}^{\infty} \frac{\arctan x}{1+x^{2}} d x\) is convergent and its value is \(\frac{\pi}{4}\)

Step 1: Check for divergent

First, check if \(\lim _{x \rightarrow \infty} \frac{\arctan x}{1+x^{2}}\) equals zero. If the limit is not zero, then the integral is automatically divergent.

Step 2: Evaluate the limit

Evaluate \(\lim _{x \rightarrow \infty} \frac{\arctan x}{1+x^{2}}\). It is well known that as x approaches infinity, \(\arctan x\) approaches \(\frac{\pi}{2}\). Thus the limit becomes \(\lim _{x \rightarrow \infty} \frac{\frac{\pi}{2}}{1+x^{2}}\), which is 0.

Step 3: Evaluate the integral

Since it's not automatically divergent, evaluate the integral \(\int_{1}^{\infty} \frac{\arctan x}{1+x^{2}} d x\). We can use the King's property also known as the Integral Comparison test which gives \(\int_{1}^{\infty} \frac{\arctan x}{1+x^{2}} d x = \int_{1}^{\infty} \frac{\arctan(\frac{1}{x})}{1+(\frac{1}{x})^{2}} d(\frac{1}{x})\). Add these two integrals which gives \(2\int_{1}^{\infty} \frac{\arctan x}{1+x^{2}} d x = \int_{0}^{1} \frac{\pi}{2} dy\), where \(y= \arctan x\). Therefore, \(\int_{1}^{\infty} \frac{\arctan x}{1+x^{2}} dx = \frac{\pi}{4}\), indicating that the integral is convergent.

Key Concepts

Improper Integrals

An improper integral is a type of integral that has either one or both limits of integration as infinity, or occurs when the integrand has an infinite discontinuity within the interval of integration. In essence, it’s an integral where the usual method of integration for intervals with finite boundaries doesn't apply directly.

Improper integrals are solved by defining them in terms of limits. For example, the integral \[\begin{equation}\int_a^{\infty} f(x) dx\end{equation}\] is defined as \[\begin{equation}\lim_{t \to \infty}\int_a^t f(x) dx,\end{equation}\] if this limit exists (i.e., is finite), then we say the integral is convergent; otherwise, it’s divergent.

To check the convergence of improper integrals, we can use different tests including comparison tests, limit tests, or seek known convergent/divergent behaviors of similar functions. This process is crucial since it informs us whether the integral assigns a finite area under a curve over an infinite interval or near an infinite discontinuity.

Integral Comparison Test

The Integral Comparison Test is a valuable method for determining the convergence or divergence of an improper integral. It involves comparing our integral of interest, often more challenging to evaluate directly, to a simpler integral whose behavior we already know.

Specifically, the test states that if \[\begin{equation}0 \leq f(x) \leq g(x)\end{equation}\] for all x in the interval \[\begin{equation}[a, \infty),\end{equation}\] and if \[\begin{equation}\int_a^{\infty} g(x) dx\end{equation}\]converges, then \[\begin{equation}\int_a^{\infty} f(x) dx\end{equation}\]also converges. Conversely, if the integral of g(x) diverges, and \[\begin{equation}\lim_{x \to \infty} \frac{f(x)}{g(x)}\end{equation}\]exists and is finite, then the integral of f(x) diverges as well.

In our exercise example, we make use of a property similar to the comparison test known as the 'King’s property', which helps in finding the value of certain symmetric integrals and eventually proving their convergence.

Limits of Integration

Limits of integration are the endpoints of the interval over which the integration is performed; they define the span of the area under the curve of the function being integrated. Proper integrals have both limits as finite numbers, while improper intgerals extend to infinity or start/end at a point of discontinuity. The limits are crucial as they dictate the behavior and possible convergence of an integral.

When dealing with improper integrals where at least one of the limits is infinite, it becomes necessary to introduce a limit process. This allows us to handle infinity in a more mathematical and precise way through the evaluation of a limit, as was shown in the example problem. Calculating the area under a curve from a point to infinity, or over a point of discontinuity, requires us to use limit notation to define the integral in a way that can be meticulously analyzed and computed.

The understanding of limits of integration is essential for solving improper integrals, as it provides the framework for defining these integrals in terms of limits, allowing us to apply tests for convergence and make meaningful interpretations of integrals in the extended real number line.