Problem 26 - Calculus for Biology and Medicine

Question

In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=\cos (3 x) $$

Step-by-Step Solution

Answer

The general antiderivative of \( f(x) = \cos(3x) \) is \( F(x) = \frac{1}{3} \sin(3x) + C \).

Step 1: Identify the Integral Formula

To find the general antiderivative of the function \( f(x) = \cos(3x) \), we first need to recognize the basic antiderivative formula for cosine. Recall that \( \int \cos(x) \, dx = \sin(x) + C \), where \( C \) is the constant of integration.

Step 2: Adjust for the Coefficient inside the Cosine Function

When the argument of the trigonometric function is more complex than just \( x \), such as \( 3x \) in this problem, we apply the reverse chain rule. If \( u = 3x \), then \( du = 3 \, dx \), making \( dx = \frac{1}{3}du \).

Step 3: Substitute and Integrate

Substituting \( u = 3x \) into the integral gives \( \int \cos(u) \frac{1}{3} \, du = \frac{1}{3} \int \cos(u) \, du \). Using the antiderivative formula, we have \( \frac{1}{3} \sin(u) + C = \frac{1}{3} \sin(3x) + C \).

Step 4: Write the General Antiderivative

The general antiderivative of the function \( f(x) = \cos(3x) \) is \( F(x) = \frac{1}{3} \sin(3x) + C \), where \( C \) is an arbitrary constant that accounts for any constant term that could be added to the function.

Key Concepts

Integral Calculus

Integral calculus is a crucial branch of calculus that focuses on the concept of integration. Here, we aim to find antiderivatives or integrals of functions. An antiderivative of a function helps us in finding the original function from its derivative. It's also known as the inverse process of differentiation.

When we talk about integrals, we refer to two broad types: definite and indefinite integrals.

• Definite Integrals: Compute the area under a curve within a specified range on the x-axis. This is equivalent to summing up infinitely thin rectangles under a curve.
• Indefinite Integrals: Concerned with finding the general form of a function that differentiates to give the integrand, without specific limits of integration. These extend the concept to include a constant of integration, which indicates that there could be multiple vertical shifts of the antiderivative function.

Given a function like the one in the exercise, \[ f(x) = \cos(3x) \]we focus on finding the indefinite integral to understand its general antiderivative.

Integral calculus forms the backbone of many scientific explorations, as it provides tools to calculate quantities where instantaneous rates of change (derivatives) are known.

Trigonometric Integration

Trigonometric integration is a specific technique within integral calculus that involves integrating functions of trigonometric forms. Often, these involve \[\sin(x), \cos(x), \tan(x),\] and other trigonometric functions. These functions are periodic and have well-defined integrals.

In our example, the function involved is \[ f(x) = \cos(3x) \]To solve this, we use a well-known basic antiderivative involving cosine:

• The integral of \( \cos(x) \) is \( \sin(x) + C \), where \( C \) is the integration constant.

However, the presence of the coefficient in our cosine function (3 in this case) means we can't apply this directly. This is where more advanced techniques come in, facilitating the modification of the integration to reflect the actual input inside the trigonometric function. This approach ties into our next core topic, the reverse chain rule.

Reverse Chain Rule

The reverse chain rule is an essential technique used in integration, especially to solve integrals involving nested or composite functions. It is the reverse counterpart of the chain rule used in differentiation.

Let’s consider a function inside another function, like \[\cos(3x)\].Here, you use this method to "reverse" the differentiation process that would have involved the chain rule.
To apply the reverse chain rule, we:

1. Identify the inner function, often set as \( u \). In this context, we choose \( u = 3x \). Then, differentiate this inner function to find \( du = 3\, dx \).
2. Re-express \( dx \) in terms of \( du \), giving us \( dx = \frac{1}{3}du \).
3. Substitute \( u \) and \( du \) back into the integral to simplify it. The integral \( \int \cos(u) \cdot \frac{1}{3} \, du \) becomes easier to compute.

The final antiderivative ends up as \( \frac{1}{3}\sin(3x) + C \), after reversing the steps of differentiation. The reverse chain rule allows for an intuitive way to tackle complex integrands by transforming them into simpler, more manageable pieces.