Problem 35 - 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. \(\int_{1}^{\infty} \frac{1}{(x+1)^{3}} d x\)

Step-by-Step Solution

Answer

The integral \(\int_{1}^{\infty} \frac{1}{(x+1)^{3}} d x\) is convergent and its value is \( \frac{1}{8} \).

Step 1: Limit calculation

First, let's calculate \( \lim _{x \rightarrow \infty} \frac{1}{(x+1)^{3}} \). If it's not equal to \(0\), then the integral is divergent. If it equals \(0\), proceed to the next step.

Step 2: Antiderivative calculation

Now, let's find the antiderivative of the function. The antiderivative of \( \frac{1}{(x+1)^{3}} \) is \( -\frac{1}{2(x+1)^{2}} \).

Step 3: Application of the Fundamental Theorem of Calculus

Apply the Fundamental Theorem of Calculus by substituting the limits of integration into the antiderivative function and subtracting. The result is \( \left[-\frac{1}{2(x+1)^{2}}\right]_{1}^{\infty} = -\frac{1}{2(\infty+1)^{2}} - \left(-\frac{1}{2(1+1)^{2}}\right) = -0 - \left(-\frac{1}{8}\right) = \frac{1}{8} \).

Key Concepts

Limit Calculation

To correctly evaluate if an integral converges or diverges, limit calculation is often the first step. Consider the given exercise where you need to determine the behavior of the integral of \( \frac{1}{{(x+1)^3}} \) as \( x \) approaches infinity. The limit \( \lim _{x \rightarrow \infty} \frac{1}{(x+1)^{3}} \) assesses the function's end-behavior.

When \( x \) becomes very large, \( (x+1)^3 \) grows rapidly, making the fraction very small, essentially nearing zero. Mathematically, for any positive power of \( x \), the limit as \( x \) approaches infinity will be zero. This property is crucial in determining the convergence of integrals, especially when working with improper integrals that have infinite limits of integration.

In this case, the limit equals zero, implying that the integral may converge, and we can proceed with further steps to evaluate it.

Antiderivative Calculation

Once limit calculation suggests the potential for convergence, the next step is antiderivative calculation. The antiderivative, also known as the indefinite integral, is the reversed process of differentiation. For the exercise at hand, finding the antiderivative of \( \frac{1}{{(x+1)^3}} \) involves applying the reverse power rule, which gives \( -\frac{1}{2(x+1)^{2}} \).

Understanding how to calculate antiderivatives is essential for evaluating definite integrals, as it sets the stage for applying the Fundamental Theorem of Calculus. This step demonstrates an important practice: when integrating functions with variables in the denominator raised to a power, we increase the power by one, multiply by the reciprocal of the new power, and change the sign.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is the bridge between differentiation and integration. It consists of two parts, with the first part providing an antiderivative for a continuous function and the second part allowing us to evaluate definite integrals. In our exercise, we use the second part: once we have the antiderivative of the function, we can evaluate the definite integral by substituting the upper and lower bounds of the integral into this antiderivative and taking the difference.

Specifically, for the integral \( \int_{1}^{\infty} \frac{1}{(x+1)^{3}} dx \), after finding the antiderivative, the Fundamental Theorem instructs us to compute \( \left[-\frac{1}{2(x+1)^{2}}\right]_{1}^{\infty} \). This is a straightforward subtraction problem involving the antiderivative's value at the upper and lower limits, highlighting the theorem's practicality in evaluating definite integrals of functions.

Improper Integral Evaluation

The concept of an improper integral is used when the interval of integration is unbounded (involves infinity) or the integrand has an infinite discontinuity. To evaluate an improper integral, we interpret it as a limit, often using the Fundamental Theorem of Calculus to find the antiderivative first and then taking the limit of the antiderivative as one of the bounds approaches infinity or a point of discontinuity.

In our exercise, the integral \( \int_{1}^{\infty} \frac{1}{(x+1)^{3}} dx \) is improper because it has an infinite upper limit. The evaluation of the improper integral involves taking the limit of the antiderivative as \( x \) approaches infinity. We note that as \( x \) grows, the term \( -\frac{1}{2(x+1)^{2}} \) becomes negligible, approaching zero. Therefore, the contributing value comes from the antiderivative evaluated at the lower limit of \( x=1 \), which results in the value of \( \frac{1}{8} \). This indicates the integral is convergent with a finite value.