Problem 31 - Calculus for Biology and Medicine

Question

Find the general antiderivative of the given function. $$ f(x)=\sec ^{2}(2 x) $$

Step-by-Step Solution

Answer

The general antiderivative is \(\frac{1}{2}\tan(2x) + C\).

Step 1: Recall the Basic Antiderivative Rule

The antiderivative of \(\sec^2(x)\) is a basic rule that states: \(\int \sec^2(x)\,dx = \tan(x) + C\), where \(C\) is the constant of integration.

Step 2: Identify and Transform the Function

Our given function is \(f(x) = \sec^2(2x)\). Notice that this involves a composite function: \(\sec^2(u)\) where \(u = 2x\).

Step 3: Use the Chain Rule for Integration

According to the chain rule for integration, if \(u = 2x\), then \(du = 2\,dx\) or \(dx = \frac{du}{2}\). Substitute these into the integral: \\[ \int \sec^2(2x) \, dx = \int \sec^2(u) \, \frac{du}{2} \] which simplifies to \(\frac{1}{2} \int \sec^2(u) \, du\).

Step 4: Integrate the Function

Integrate using the basic antiderivative rule from Step 1: \\[ \frac{1}{2} \int \sec^2(u) \, du = \frac{1}{2} \tan(u) + C \]

Step 5: Substitute Back for x

Replace \(u\) back with \(2x\) to express the antiderivative in terms of \(x\): \\[ \frac{1}{2} \tan(2x) + C \]

Step 6: Write the Final Answer

Thus, the general antiderivative of \(\sec^2(2x)\) is \(\frac{1}{2}\tan(2x) + C\).

Key Concepts

Chain Rule

The chain rule is a fundamental concept in both differentiation and integration when dealing with composite functions. In the context of integration, the chain rule helps us understand how to handle functions composed of other functions.
For example, when you see a function like \( \sec^2(2x) \), it's not a simple function. Instead, it is composed of \( \sec^2(u) \) where \( u = 2x \). This is actually multiple layers deep, with \( 2x \) being the inner function. To integrate a function like this, we can use a modification of the derivative chain rule, sometimes called "u-substitution." Here's how it works:

• Identify the inner function: \( u = 2x \).
• Differentiate this inner function: \( du = 2 \, dx \).
• Solve for \( dx \) in terms of \( du \): \( dx = \frac{du}{2} \).

By substituting these expressions into your integral, it transforms into a simpler form that's easier to integrate.

Integration Techniques

Integration techniques are the various methods we use to solve integrals. Each technique has its own set of rules and scenarios where it applies best. Understanding which technique to use lies at the heart of successful integration.In the exercise provided, the technique we used involved both recognizing a basic antiderivative and applying a substitution method.

• First, recall the basic antiderivative rule for \( \sec^2(x) \): \( \int \sec^2(x) \; dx = \tan(x) + C \).
• Next, notice the substitution technique. By expressing \( dx \) as \( \frac{du}{2} \) (derived from our chain rule step), the integral of the composite function becomes manageable: \( \frac{1}{2} \int \sec^2(u) \; du \).
• Finally, apply the basic rule: \( \frac{1}{2} \tan(u) + C \), and substitute back for the original variable to complete the integration.

This example perfectly demonstrates how different techniques can work together to solve complex integrals.

Constant of Integration

The constant of integration, denoted as \( C \), is an essential part of indefinite integrals. When we find an antiderivative, we often add this constant because integration can yield a whole family of functions.For example, if \( F(x) \) is an antiderivative of \( f(x) \), so is \( F(x) + C \). Why is it so? Because the derivative of a constant is zero, thus does not change the result when differentiating:

• Imagine calculating the derivative of \( F(x) + C \): you simply get \( f(x) \).
• Since any constant \( C \) differentiates to zero, the expression \( F(x) + C \) encompasses all potential antiderivatives of \( f(x) \).

Always remember to include \( C \) when performing indefinite integrals as it ensures your solution is general and complete.