Problem 23 - Calculus for Biology and Medicine

Question

Determine whether each integral is convergent. If the integral is convergent, compute its value. $$ \int_{0}^{\infty} \frac{1}{\sqrt{x+1}} d x $$

Step-by-Step Solution

Answer

The integral converges, and its value is 2.

Step 1: Analyze the Integral for Congruence

We need to check if \( \int_{0}^{\infty} \frac{1}{\sqrt{x+1}} \, dx \) is convergent. The integral is improper because its upper limit is \( \infty \). If \( f(x) = \frac{1}{\sqrt{x+1}} \) satisfies \( \lim_{x \to \infty} f(x) = 0 \), we can find convergence by calculating the limit of the antiderivative as \( x \to \infty \).

Key Concepts

Convergence of Integrals

Improper integrals, like the one in our exercise, extend over an infinite boundary. This raises the question of convergence. For an integral like \( \int_{0}^{\infty} \frac{1}{\sqrt{x+1}} \, dx \), convergence means that, as we integrate from 0 to infinity, the sum of all these tiny slices of area under the curve remains finite. To determine convergence, observe the behavior of the function - \( \frac{1}{\sqrt{x+1}} \) - as \( x \) approaches infinity:

• We aim to understand if the function approaches zero fast enough so that its integral results in a finite value.
• We can achieve this by substituting a limit into our integral since this is a classic method for evaluating improper integrals.

When the limit of the function correctly approaches zero as \( x \to \infty \), it hints toward the potential convergence of the integral. Once we find convergence, the next step is finding the exact value of the integral. This is achieved through antiderivatives and limits.

Antiderivatives

An antiderivative is a function whose derivative is the given function. To evaluate the integral \( \int_{0}^{\infty} \frac{1}{\sqrt{x+1}} \, dx \), we first need to find its antiderivative. Consider a substitution or a known antiderivative form to simplify the function:

• The goal here is to transform the function into a simple expression that is easy to integrate.
• For \( \frac{1}{\sqrt{x+1}} \), an antiderivative is \( 2\sqrt{x+1} \).
• Knowing this, we can further substitute limits and solve.

Finding the antiderivative is a crucial step to solving integrals, as it represents a function where, if differentiated, returns the original function. This facilitates the computation of definite integrals over desired intervals, particularly where the limits are evaluated.

Limit of a Function

Limits are essential in dealing with improper integrals. When an integral has an infinite upper bound, such as \( \int_{0}^{\infty} \frac{1}{\sqrt{x+1}} \, dx \), we look towards limits to assess the behavior at infinity.For our integral:

• The function \( \frac{1}{\sqrt{x+1}} \) slowly approaches zero as \( x \to \infty \).
• In practical terms, this means we're adding progressively smaller quantities, suggesting finiteness.
• By evaluating \( \lim_{b \to \infty} \int_{0}^{b} \frac{1}{\sqrt{x+1}} \, dx \), solving this limit using antiderivatives, confirms the convergence of the integral.

The limit provides a tool to manage infinite behaviors by approaching large values and assessing their impact on function sums. This approach complements the understanding and solutions of improper integrals effectively.