Problem 18 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Compute the following integrals. $$ \int_{0}^{\ln 5} \frac{3 e^{x}}{\sqrt{e^{x}+4}} d x $$

Step-by-Step Solution

Answer

The value of the integral is \( 3 \cdot 2 \cdot (\arcsin(2) - \arcsin(\frac{10}{9})) \)

Step 1: Variable substitution

Let's set \( u = e^{x} + 4 \). Thus, \( du = e^{x} dx \) which implies \( dx = du / e^{x} \). The new limits of integration will be \( u(0) = e^{0} + 4 = 5 \) and \( u(\ln 5) = 5 + 4 = 9 \). Substituting these into the integral we get:\[ \int_{5}^{9} \frac{3}{\sqrt{u}} \frac{du}{u - 4}\]

Step 2: Simplifying the Integral

The integral can be simplified further as:\[ \int_{5}^{9} \frac{3}{\sqrt{u} \cdot (u - 4)} du = 3 \int_5^9 \frac{du}{\sqrt{u} \cdot (u - 4)}\]

Step 3: Calculate the Integral

We can solve this integral by using the direct formula for \( \int \frac{dx}{\sqrt{x(a-x)}} = 2 \arcsin(\frac{2x}{a}) \), which gives:\[= 3 \cdot [2 \arcsin(\frac{2u}{9})]_5^9 = 3 \cdot 2 \cdot (\arcsin(2) - \arcsin(\frac{10}{9}))\]

Key Concepts

Integration Techniques

In the world of single variable calculus, solving integrals can sometimes feel like solving a puzzle. When approaching integrals, there are a variety of techniques that can be employed to find solutions.
One popular method is substitution, which is particularly useful when dealing with complex or non-standard integrands. This method simplifies the terms by introducing a new variable. Another technique is partial fractions, which works well when dealing with rational functions. In other situations, trigonometric identities or integration by parts might be more helpful.
Each technique has its own strengths and is chosen based on the type of integral at hand. By using these methods, integrals that initially seem challenging become more straightforward and manageable. It's like having a toolbox, and with practice, you learn which tool fits which problem the best.

Substitution Method

The substitution method is a fundamental technique in integration that simplifies the process by transforming the integral into a more familiar form. This is done by changing the variable of integration to ease the calculation.
In the given exercise, a substitution was made by letting \( u = e^{x} + 4 \). This change of variables transforms the original integral into something simpler, making it easier to compute.
Here's how it works:

• Choose a new variable \( u \).
• Express \( dx \) in terms of \( du \) and the derivative of the substitution.
• Replace the original limits of integration with the new ones for \( u \).
• After integration, substitute back to the original variable if necessary.

The primary goal is to make the integrand simpler, and this technique is particularly useful when the integrand is a composite function, as differentials become more manageable.

Definite Integrals

Definite integrals are an essential concept in calculus, representing the accumulation of quantities and often giving the area under a curve. With definite integrals, we calculate the integral within specified limits, denoted as \( a \) to \( b \).
In our problem, the limits were from 0 to \( \ln 5 \). These boundaries define the region over which the function is integrated. When performing integration using substitution or any other method, these limits change accordingly with the new variable.
The crucial steps include:

• Setting up the integral with new limits that correspond to the substituted variable.
• Solving the integral within these limits.
• Applying the Fundamental Theorem of Calculus, which states that if \( F \) is an antiderivative of \( f \), then the definite integral of \( f \) over \( [a, b] \) is \( F(b) - F(a) \).

Understanding definite integrals helps in finding precise values for areas, total quantities, or solving physics problems that involve continuous change.