Problem 18 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The general antiderivative of \( f(x) = \frac{1}{1+3x} \) is \( \frac{1}{3} \ln |1 + 3x| + C \).

Step 1: Recognize the Form

The function given is of the form \( f(x) = \frac{1}{1+3x} \). Notice that this function resembles the derivative form \( \frac{d}{dx}\left(\ln |u| \right) = \frac{1}{u} \cdot \frac{du}{dx} \).

Step 2: Identify Substitution

Let \( u = 1 + 3x \). This means that \( du = 3 \, dx \) or equivalently, \( dx = \frac{du}{3} \).

Step 3: Perform Substitution

Substitute \( u \) and \( dx \) into the integral: \[ \int \frac{1}{u} \cdot \frac{du}{3} = \frac{1}{3} \int \frac{1}{u} \, du \].

Step 4: Integrate

The integral of \( \frac{1}{u} \, du \) is \( \ln |u| + C \), where \( C \) is the constant of integration. Therefore, \[ \frac{1}{3} \int \frac{1}{u} \, du = \frac{1}{3} \ln |u| + C \].

Step 5: Substitute Back

Replace \( u \) with \( 1 + 3x \) to write the antiderivative in terms of \( x \): \[ \frac{1}{3} \ln |1 + 3x| + C \].

Key Concepts

Substitution Method

Understanding the substitution method is vital when dealing with integrals that fit certain forms, like the one we have here. This technique simplifies the integration process by allowing us to replace a complex part of the integrand with a single variable. In this exercise, we notice that the denominator, \(1 + 3x\), can be treated as a single unit.

• We start by letting \(u = 1 + 3x\).
• Consequently, the derivative \(du = 3\,dx\) helps us express \(dx\) in terms of \(du\), resulting in \(dx = \frac{du}{3}\).

This substitution transforms the integral into a simpler form, making it easier to handle. This technique is especially useful when the integrand includes composite functions, providing a straightforward path to integration.

Integration

Integration is the process of finding the antiderivative of a function. In this context, we've taken advantage of the substitution method to rewrite the integral in a simpler form: \(\int \frac{1}{u}\, du\). This integral is one of the basic results often memorized due to its frequent occurrence:

• The antiderivative of \(\frac{1}{u}\) is \(\ln |u|\).
• Therefore, integrating \(\int \frac{1}{u}\, du\) gives us \(\ln |u| + C\), where \(C\) is the constant of integration.

The integration step is crucial because it gives us the general form of the original function's antiderivative. Transforming back the substituted variable \(u\) to terms of \(x\) completes the integration process.

Constant of Integration

Whenever we find an indefinite integral, like in this problem, we must add a constant of integration, denoted as \(C\). But why is this constant necessary?

• The reason is that when we take the derivative, the constant disappears. Hence, many functions have the same derivative.
• By adding \(C\), we represent the entire family of antiderivatives, all of which differentiate into the original function.

This constant of integration assures completeness in our solution set. For instance, any shifted version of the antiderivative function still aligns with the derivative of the original expression. That's the beauty and necessity of \(C\) in every indefinite integration.