Problem 48 - Calculus: Early Transcendental Functions

Question

Make the indicated substitution for an unspecified function \(f(x)\). $$u=\sqrt{x} \text { for } \int_{0}^{4} \frac{f(\sqrt{x})}{\sqrt{x}} d x$$

Step-by-Step Solution

Answer

The integral \(\int_{0}^{4} \frac{f(\sqrt{x})}{\sqrt{x}} dx\), upon performing the substitution \(u=\sqrt{x}\), results in the integral \(\int_{0}^{2} 2f(x) dx\).

Step 1: Perform the Substitution

Start by substituting \(u = \sqrt{x}\) into the integral and understand how this changes the limits of integration and the differential \(dx\). Given that \(u = \sqrt{x}\), we can square both sides to get \(x = u^2\). The differential \(dx\) can now be found by differentiating \(x = u^2\) with respect to \(u\), which gives \(dx = 2u du\). The limits of integration should change to reflect the values of \(u = \sqrt{x}\). When \(x = 0\), \(u = \sqrt{0} = 0\), and when \(x = 4\), \(u = \sqrt{4} = 2\). Therefore, our variable-substituted integral becomes: \(\int_{0}^{2} \frac{f(u)}{u} \cdot 2u du\).

Step 2: Simplify the Integral

Upon substitution, the integrand simplifies since the \(u\) in the denominator cancels with the \(u\) in the numerator. This simplification yields a new, simpler integral: \(\int_{0}^{2} 2f(u) du\).

Step 3: Resulting Integral

The resulting integral is thus \(\int_{0}^{2} 2f(u) du\). However, since the original problem gives the function \(f(x)\) as unspecified, the integral cannot be evaluated further. The notation often returns to the original variable, so the answer is usually expressed as \(\int_{0}^{2} 2f(x) dx\).

Key Concepts

Indefinite Integral

At the heart of solving calculus problems lies the indefinite integral. It is the gateway to finding the function whose derivative is given. Unlike definite integrals, which calculate the area under the curve between two points, an indefinite integral does not have specified limits of integration and includes a constant of integration, often denoted as 'C'.

Understanding the indefinite integral is crucial for problems involving antiderivatives. When dealing with functions like \( f(x) \), the indefinite integral is represented as \( \int f(x) \, dx \), and the goal is to determine the general form of the original function before differentiation. Since indefinite integrals lack boundary values, they are more about the process than the specific result.

Integration Techniques

Integration is not always straightforward, especially for complex functions. That's where integration techniques come into play. They are strategies to simplify integrals and make them more manageable. One common method is u-substitution, which we see in the sample exercise. Other techniques include integration by parts, partial fraction decomposition, trigonometric substitution, and more.

Each technique has its criteria and applications. Selecting the right method often depends on recognizing patterns or forms within the integrand that match well with a particular technique. Mastery of these methods allows students to tackle a wider range of integration problems.

Calculus

Calculus is a branch of mathematics that focuses on the study of change and motion. It is divided primarily into two fields: differential calculus concerning rates of change and slopes of curves, and integral calculus, which deals with accumulation of quantities and areas under or between curves.

In essence, where differential calculus cuts something into small pieces to find how it changes, integral calculus joins (integrates) the pieces together to find out how much there is. The exercise given involves integral calculus, specifically aimed at finding a function through its derivative and the technique of u-substitution to simplify the process.

Substitution Method

The substitution method, also known as u-substitution, is an approach utilized to simplify integrals. It is particularly useful when an integral contains a function and its derivative. In u-substitution, we select a part of the integrand to replace with a variable 'u', which simplifies the integral. After substitution, the integral is usually easier to evaluate.

Implementing u-substitution involves a few steps: choosing the appropriate section of the integrand as 'u', differentiating 'u' to find 'du', and substituting both 'u' and 'du' back into the integral. After solving the integral with respect to 'u', we typically substitute back into the original variable, as in the step-by-step solution provided.