Problem 11 - Calculus for Biology and Medicine

Question

In Problems 1-16, evaluate each indefinite integral by making the given substitution. $$ \int x e^{x^{2} / 2} d x, \text { with } u=x^{2} / 2 $$

Step-by-Step Solution

Answer

The integral is \( e^{x^2/2} + C \).

Step 1: Identify the substitution

We are given the substitution \( u = \frac{x^2}{2} \). This means that our task is to rewrite parts of the integral in terms of \( u \). First, differentiate \( u \) with respect to \( x \).

Step 2: Differentiate to find du

Differentiate \( u \) to find \( du \). Since \( u = \frac{x^2}{2} \), the derivative is \( \frac{d}{dx}(\frac{x^2}{2}) = x \). Therefore, \( du = x \, dx \).

Step 3: Substitute in the integral

Now substitute \( u \) and \( du \) into the integral. The integral \( \int x e^{x^2/2} dx \) becomes \( \int e^u \, du \). This is because \( x \cdot dx = du \) and \( e^{x^2/2} = e^u \).

Step 4: Integrate with respect to u

The integral \( \int e^u \, du \) is straightforward. It evaluates to \( e^u + C \), where \( C \) is the constant of integration.

Step 5: Back-substitute for x

Return to the original variable by substituting \( u = \frac{x^2}{2} \) back into the solution. This gives us \( e^{x^2/2} + C \) as the evaluated integral.

Key Concepts

Indefinite Integral

An indefinite integral, often simply called an "integral," represents a family of functions whose derivative is the integrand. Think of it as reversing the process of differentiation. When you evaluate an indefinite integral, you are essentially looking for a function whose derivative gives you the original function or expression inside the integral.

When we denote an indefinite integral, we use the integral sign followed by the function and the differential (like \( \int f(x) \, dx \)). Solving this answers the question: "What was differentiated to result in \( f(x) \)?" One important thing to remember is that indefinite integrals include a constant of integration, denoted as \( C \), because differentiation of a constant results in zero, and thus any constant could be part of our original function.

Differentiation

Differentiation is one of the core operations in calculus. It involves finding the derivative of a function, which is a measure of how the function's output value changes as its input changes. The derivative is essentially the rate of change or the slope of the function at any given point.

In the original problem, we differentiated the substitution \( u = \frac{x^2}{2} \) to find \( du \). Calculating the derivative helps in understanding the changes in \( u \) with respect to changes in \( x \), which is crucial for rewriting our integral in terms of \( u \). This rewriting makes the integral easier to evaluate, as many complex expressions can be transformed into simpler integrals by substitution.

Calculus

Calculus is a branch of mathematics focused on change and motion. It consists of two main parts: differentiation and integration, which are, in a way, opposites of each other.

Differentiation deals with finding the rate at which things change, while integration concerns adding up small pieces to find the whole, such as areas under curves or the accumulated change over time. These concepts are foundational for science and engineering, allowing us to model the real world, optimize processes, and more. In our exercise, both differentiation and integration were used. We used differentiation to find \( du \) and integration to evaluate the integral \( \int e^u \, du \). Together, these methods make calculus a powerful tool for solving problems involving change.

Exponential Function

The exponential function is a type of mathematical function denoted as \( e^x \), where \( e \) is Euler's number, approximately 2.71828. This function is unique in that it is its own derivative and integral, making it an especially simple function to work with in calculus when it comes to integration and differentiation.

In the exercise, the integrand \( e^{x^2/2} \) involves the exponential function. After substitution, it transforms into the simpler \( e^u \). The ease of integrating \( e^u \) is due to the fact that integration results in \( e^u + C \), with the exponential function maintaining its straightforward form. This characteristic often simplifies problems involving exponential functions in calculus, allowing us to focus on other challenges such as determining the appropriate substitutions.