Problem 40 - Calculus for Biology and Medicine

Question

In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=5 e^{3 x}-\sec ^{2}(x-3) $$

Step-by-Step Solution

Answer

The general antiderivative is \( \frac{5}{3}e^{3x} - \tan(x-3) + C \).

Step 1: Identify the components of the function

The given function is \( f(x) = 5e^{3x} - \sec^2(x-3) \). This function is composed of two separate terms: \( 5e^{3x} \) and \( -\sec^2(x-3) \). Each term can be integrated separately.

Step 2: Find the antiderivative of the first term

The first term is \( 5e^{3x} \). The antiderivative of \( e^{ax} \) is \( \frac{1}{a}e^{ax} \). So, the antiderivative of \( 5e^{3x} \) is \( \frac{5}{3}e^{3x} \).

Step 3: Find the antiderivative of the second term

The second term is \( -\sec^2(x-3) \). The antiderivative of \( \sec^2(u) \) is \( \tan(u) \). Applying the chain rule, the antiderivative of \( \sec^2(x-3) \) is \( \tan(x-3) \). Therefore, the antiderivative of \( -\sec^2(x-3) \) is \( -\tan(x-3) \).

Step 4: Combine the antiderivatives

Combine the antiderivatives of each term to form the general antiderivative of \( f(x) \). Add a constant of integration, \( C \), since we don't have limits of integration. Therefore, the general antiderivative of \( f(x) \) is \( \frac{5}{3}e^{3x} - \tan(x-3) + C \).

Key Concepts

Calculus

Calculus is a branch of mathematics that helps us understand changes. It's the study of how things move and where they are going. Unlike basic arithmetic or algebra, calculus introduces us to concepts like limits, derivatives, and integrals, which become essential in understanding more complex problems. It plays a crucial role in physics, engineering, economics, and even biology.

With calculus, we can find the rate at which things change and accumulate over time. It works with both very small numbers (infinitesimals) and very large numbers. The two major concepts of calculus are differentiation and integration, which we can think of as operations that are the reverse of each other. Differentiation is about finding the rate of change or the slope of a curve, and integration is about finding the total accumulation or the area under a curve.

An essential idea in calculus is the antiderivative. This is a function that reverses the process of differentiation. Thus, finding an antiderivative is akin to solving a backward puzzle, where you know the rate at which something changes, but you want to trace back to the original condition.

Integration

Integration is like a camera capturing every tiny bit of a picture to develop the whole image. In mathematics, it is the skill of finding areas, volumes, central points, and more. It involves summing up infinitesimal parts to calculate the whole thing.

There are two main types of integration:

• Indefinite Integration: This gives us the general form of an integral with a constant added. It doesn't have specific limits, so it represents a family of functions. For example, finding the antiderivative of a given function is an indefinite integration task.
• Definite Integration: This computes the value with specified limits of integration, providing specific number outcomes, such as areas under curves or exact figures for a given range.

In our exercise, we're performing indefinite integration to find the general antiderivative. This involves working with each part of the function separately, finding the integral, and then combining them with a constant of integration, usually represented by the symbol C. Integration can seem tricky at first, but it becomes clearer with practice and understanding of its basic rules.

Chain Rule

The chain rule is a powerful tool in calculus for dealing with compositions of functions. When you have a function inside another function, the chain rule helps find the derivative of this composite effectively. It’s a bit like peeling an onion layer by layer from the outside in.

In the context of integration, we often have to "reverse" this idea. When dealing with functions like \(\sec^2(x-3)\), the chain rule in reverse comes into play. Here’s how it works:

• First, identify the "inner" function (like \(x-3\) in the example).
• Second, take the derivative or integral of the "outer" function while accounting for the inner function's behavior.

In our example, the chain rule allows us to effectively "undo" the differentiation done before, getting us back to the original condition. This step crucially shows the essence and power of calculus, allowing for exploration of complex systems by deconstructing them into simpler parts.