Problem 43 - Calculus for Biology and Medicine

Question

Assume that a is a positive constant. Find the general antiderivative of the given function. $$ f(x)=\frac{1}{a x+3} $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Problem

We need to find the general antiderivative of the function \( f(x) = \frac{1}{a x + 3} \). An antiderivative is a function whose derivative equals the given function. In this case, it involves integration.

Step 2: Set Up the Integral

To find the antiderivative, we need to set up the integral: \( \int \frac{1}{a x + 3} \, dx \). This integral resembles the standard form \( \int \frac{1}{u} \, du \).

Step 3: Substitution

Let \( u = a x + 3 \), then \( du = a \, dx \). Solving for \( dx \), we get \( dx = \frac{du}{a} \). Substitute these into the integral, which changes to \( \int \frac{1}{u} \cdot \frac{du}{a} \).

Step 4: Simplify the Integral

The integral becomes \( \frac{1}{a} \int \frac{1}{u} \, du \). The integral of \( \frac{1}{u} \, du \) is a standard integral, which is \( \ln|u| + C \), where \( C \) is the integration constant.

Step 5: Substitute Back

Substitute \( u = a x + 3 \) back into the integrated result. This gives us \( \frac{1}{a} (\ln|a x + 3| + C) \). Simplifying, the general antiderivative is \( \frac{1}{a} \ln|a x + 3| + C \).

Key Concepts

Antiderivative

When tackling integrals, one of the first concepts you will encounter is the antiderivative. An antiderivative is a function whose derivative is the given function. Knowing this helps us reverse the process of differentiation, essentially 'undoing' a derivative. For the function \( f(x) = \frac{1}{a x + 3} \), the task is to find such a function, because its derivative should give us back the original \( f(x) \).
To find the antiderivative, we integrate the function. Though sometimes these integrals may seem complex at first glance, recognizing specific forms and utilizing methods like substitution (which we'll cover next) allows you to work through them systematically. Understanding antiderivatives is vital in calculus and helps solve real-world problems involving cumulative quantities, such as total distance from velocity functions.

Substitution Method

The substitution method, also known as u-substitution, is one of the most powerful techniques in integration. It simplifies finding the antiderivative by changing variables to transform the integral into a form that is easier to handle. In our example, with \( f(x) = \frac{1}{a x + 3} \), substitution is key as the function closely resembles \( \int \frac{1}{u} \, du \), a much simpler integral to solve.
By setting \( u = a x + 3 \), we transformed the original variable.
Next, derive \( u \) with respect to \( x \) to find \( du = a \, dx \). Solving for \( dx \) gives \( dx = \frac{du}{a} \). Substituting these into the integral, we arrive at \( \int \frac{1}{u} \cdot \frac{du}{a} = \frac{1}{a} \int \frac{1}{u} \, du \). Notice how substitution can turn a complex problem into a straightforward one.

Calculus for Biology and Medicine

While the intricacies of calculus can seem abstract, its applications are far-reaching, notably in fields like biology and medicine. Understanding antiderivatives and integration in calculus becomes crucial when analyzing biological systems' dynamics, such as population growth models, medication dosage calculations, and blood flow rates in physiology.
For example, integrating a rate of change (like medication decay in the bloodstream) allows researchers and medical professionals to predict total exposure over time and adjust treatment plans accordingly. By understanding how these mathematical frameworks apply to biological contexts, students can gain a deeper appreciation for calculus's practical utility. This knowledge empowers decision-making and critical analysis in medical research and healthcare solutions.

Constant of Integration

The constant of integration, denoted as \( C \), is an integral part of finding antiderivatives. Whenever you integrate, you must include \( C \) because integration is an inverse operation of differentiation, which loses constant values.
This constant accounts for all possible vertical shifts of the antiderivative on a graph.

• The antiderivative represents a family of functions, all differing by a constant amount.
• Without \( C \), we would miss a critical aspect of functions arising from different initial conditions.

For instance, when returning to the integral \( \int \frac{1}{u} \, du \), its antiderivative is \( \ln|u| + C \), representing an infinite set of solutions. Only additional conditions, such as boundary values, can determine the specific value of \( C \). Understanding the role of the constant of integration helps in accurately modeling and solving differential equations in both theoretical and practical scenarios.