Problem 32 - Calculus for Biology and Medicine

Question

Find the general antiderivative of the given function. $$ f(x)=\sec ^{2}(4 x) $$

Step-by-Step Solution

Answer

The general antiderivative of \( f(x) = \sec^2(4x) \) is \( \frac{1}{4} \tan(4x) + C \).

Step 1: Recognize the Function

The function we need to find the general antiderivative for is \( f(x) = \sec^2(4x) \). We recognize that this is similar to the derivative of the tangent function, since \( \frac{d}{dx}[\tan(x)] = \sec^2(x) \).

Step 2: Consider U-substitution

Because the function involves \( 4x \) instead of just \( x \), we will use a substitution method. Let \( u = 4x \), which implies \( du = 4 \, dx \). Then, \( dx = \frac{1}{4} \, du \).

Step 3: Rewrite the Integral

Rewrite the integral in terms of \( u \) as follows: \( \int \sec^2(4x) \, dx = \int \sec^2(u) \cdot \frac{1}{4} \, du \). This simplifies to \( \frac{1}{4} \int \sec^2(u) \, du \).

Step 4: Find the Antiderivative

The antiderivative of \( \sec^2(u) \) is \( \tan(u) + C \), where \( C \) is the constant of integration. Thus, \( \int \sec^2(u) \ du = \tan(u) + C \).

Step 5: Substitute back to x

Substitute \( u = 4x \) back into the antiderivative expression: \( \frac{1}{4} \int \sec^2(u) \, du = \frac{1}{4}(\tan(u) + C) = \frac{1}{4}(\tan(4x) + C) \).

Step 6: Write Final Expression

Distribute the \( \frac{1}{4} \) to obtain the final expression for the general antiderivative: \( \frac{1}{4} \tan(4x) + C \).

Key Concepts

U-substitution

U-substitution is a powerful technique used in calculus to simplify the process of integration. It involves changing the variables of integration from one set to another, which makes a complex integral easier to evaluate. In our specific example, we needed to find the antiderivative of the function \( f(x) = \sec^2(4x) \). This function involves the composite argument \( 4x \), instead of just \( x \), which hints that u-substitution may be a handy approach.

Here's how it works:

• Identify the inner function: In the expression \( \sec^2(4x) \), the inner function is \( 4x \).
• Choose a substitution: Set \( u = 4x \) so that \( du = 4 \, dx \), which leads to \( dx = \frac{1}{4} \, du \).
• Rewrite the integral: By substituting these into the integral, we transform it into a simpler form.

The purpose of u-substitution is to make the integral resemble a standard form. Once transformed, it can often be found in tables of integrals or through direct computation.

Integration

Integration is the process of finding an antiderivative or integral, which is essentially the opposite operation of differentiation. In the context of our function \( f(x) = \sec^2(4x) \), integration aims to find a function whose derivative is \( \sec^2(4x) \).

After substitution, we get an integral in terms of \( u \):

• \( \int \sec^2(u) \cdot \frac{1}{4} \, du \) simplifies to \( \frac{1}{4} \int \sec^2(u) \, du \).
• This transforms a more complicated integral into something straightforward: \( \sec^2(u) \).

The integral of \( \sec^2(u) \) is \( \tan(u) + C \), where \( C \) is the constant of integration. This is found in integral tables or can be derived by understanding that the derivative of \( \tan(u) \) is \( \sec^2(u) \). Once you solve the integral in terms of \( u \), substitute back the original variable to finish the integration process.

Trigonometric Functions

Trigonometric functions play a crucial role in calculus, especially when working with problems involving integrals and derivatives. In this case, understanding the derivative relationships of trigonometric functions is essential.

For example:

• The derivative of \( \tan(x) \) is \( \sec^2(x) \), which means while integrating \( \sec^2(x) \), we revert to the function whose derivative it represents, \( \tan(x) \).
• Recognizing such relationships is key when performing tasks like finding antiderivatives and solving integrals.

In the original problem, once we transformed to \( \sec^2(u) \), we could use the integral of \( \sec^2(u) \) to easily identify the antiderivative. This understanding of how differentiation and integration work with trigonometric functions allows us to reverse-engineer derivatives to find antiderivatives successfully.