Problem 29 - 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. \(\int_{1}^{\infty} \frac{x}{\sqrt{3+x^{2}}} d x\)

Step-by-Step Solution

Answer

By carrying out the above steps, we find that \( \lim _{u \rightarrow \infty} 2*\sqrt{u} = \infty \), which is not finite. Therefore, given integral is divergent.

Step 1: Apply integration by substitution

Let \( u = 3 + x^2 \), then the differential du = 2x dx. The integral changes from \( \int_{1}^{\infty} \frac{x}{\sqrt{3+x^{2}}} dx \) to \( 0.5 * \int_{4}^{\infty} \frac{1}{\sqrt{u}} du \)

Step 2: Evaluate the resulting integral

The integral is of standard form and simplifies to \( 0.5 * \int_{a}^{\infty} u^{-1/2} du \), which evaluates to \(0.5 * [2*\sqrt{u}]_{4}^{\infty}\)

Step 3: Substitute the limits of the integral

To determine whether the integral converges or diverges, substitute the limits of the integral into the evaluated integral function: \(f(\infty) - f(4)\)

Step 4: Test for divergence

Calculate the value of \( \lim _{u \rightarrow \infty} \), if the result isn’t finite, then the given integral is Divergent.

Key Concepts

Convergence and Divergence

Understanding whether an integral is convergent or divergent is essential when dealing with improper integrals. To determine this, we evaluate the behavior of the integral as the variable approaches a point, often infinity. In this context:

• Convergence: When the integral approaches a finite value, meaning it sums up to a finite amount.
• Divergence: When the integral either increases without bound or cannot be summed up to a finite amount.

For the integral \( \int_{1}^{\infty} \frac{x}{\sqrt{3+x^{2}}} dx \), we need to check its behavior as \( x \) approaches infinity. If the limit is not finite, then the integral diverges. We apply transformation techniques to make this determination feasible.

Integration by Substitution

Integration by substitution is a technique that helps simplify complex integrals by making a clever change of variables. This is especially useful for handling improper integrals that might initially seem too complicated to solve.In our problem, we set \( u = 3 + x^2 \). This substitution transforms the integral from a function of \( x \) to a function of \( u \). The main steps involve:

• Finding the derivative: \( du = 2x \, dx \).
• Rewriting the integral: \( \int_{1}^{\infty} \frac{x}{\sqrt{3+x^{2}}} dx \) becomes \( 0.5 * \int_{4}^{\infty} \frac{1}{\sqrt{u}} du \).

This substitution simplifies the integral, making it easier to evaluate whether it converges or diverges.

Limits at Infinity

To determine the convergence of an improper integral, evaluating the limit of the function as it approaches infinity is crucial.For the component \( 0.5 * [2*\sqrt{u}]_{4}^{\infty} \), we explore its behavior as \( u \to \infty \). If the resultant value is finite, the integral converges; otherwise, it diverges.In our specific case, as \( u \) increases, \( \sqrt{u} \) grows, leading to the conclusion that \( \lim_{u \to \infty} \sqrt{u} \) is infinite. Hence:

• The integral diverges because the evaluated limit at infinity is infinite.
• The overall value increases without bound, confirming the divergence.

Testing the limits at infinity is vital for ascertaining whether improper integrals like this are convergent or divergent.