Problem 26 - Calculus for Biology and Medicine

Question

Use substitution to evaluate the indefinite integrals. $$ \int \cos x e^{\sin x} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \\( e^{\sin x} + C \\\).

Step 1: Identify the Substitution

We notice that the expression inside the exponential function is \( \sin x \). We can use substitution by letting \( u = \sin x \). This means that \( \frac{du}{dx} = \cos x \), hence \( du = \cos x \, dx \).

Step 2: Rewrite the Integral Using Substitution

Substitute \( u = \sin x \) and \( du = \cos x \, dx \) into the integral. The integral becomes \( \int e^{u} \, du. \).

Step 3: Integrate with Respect to u

The integral of \( e^{u} \) with respect to \( u \) is simply \( e^{u} + C \), where \( C \) is the constant of integration.

Step 4: Substitute Back to x

Recall the substitution \( u = \sin x \). Substitute back to get \( e^{\sin x} + C \) as the result of the original integral.

Key Concepts

Understanding Indefinite Integrals

An integral can be thought of as the mathematical tool to find accumulated quantities. When we talk about indefinite integrals, we refer to integrals that do not have specified limits or boundaries. So how is an indefinite integral different? It results in a general function plus a constant of integration. This constant, denoted as \( C \), represents all the possible vertical shifts of the function's antiderivative.

Indefinite integrals play a crucial role in calculus as they help to reverse the process of differentiation. Thus, if given a derivative, integrating it will lead to the original function it was derived from, subject to the constant of integration. This reverse operation allows us to reconstruct the original function.

• Represents an antiderivative.
• Has no boundaries or limits of integration.
• Includes a constant of integration \( C \).

Understanding how to rewrite functions as indefinite integrals and subsequently solve them strengthens one's ability in solving complex calculus problems.

Exploring Exponential Functions

Exponential functions are an essential element of calculus and mathematics in general. They are defined as functions with a constant base raised to the power of a variable, typically shown as \( e^x \). The constant \( e \) is approximately 2.718 and is known as Euler's number.

Exponential functions are smooth, continuous, and always take positive values, which makes them significantly important in growth problems, such as population models or financial growth computations. This type of function grows rapidly as its variable increases.

• Base is a constant, used predominantly with \( e = 2.718 \).
• Found in various real-world applications, including growth processes.
• Grows or decays exponentially depending on the power.

In the context of indefinite integrals, processes like integration by substitution, often involve these exponential functions, primarily because of how cleanly they can transform during integration.

Integration by Substitution: A Fundamental Calculus Technique

Integration by substitution is a powerful technique used in calculus to simplify integration, especially when dealing with complex functions. It relies on transforming the original integral into a simpler form for easier computation.

To efficiently use this technique, identify a substitution, typically represented by a different variable, that can reduce complexity. In this process, a differential substitution is made, turning the integral into an easier-to-handle function.

• Simplifies integrals by changing variables.
• Typically involves a substitution that replaces a complex part.
• Utilizes a differential substitution, making the problem easier.

Let's take a simple example of substitution: In the given problem, we identified \( u = \sin x \), simplifying the integral concerning \( e^u \). Successfully applying substitution often leads to a manageable expression, which can be integrated easily, ultimately rewinding the substitution to return to the original variable context.