Problem 9 - Calculus Early Transcendentals

Question

Use a table of integrals to determine the following indefinite integrals. $$\int \frac{3 u}{2 u+7} d u$$

Step-by-Step Solution

Answer

Question: Find the indefinite integral of the rational function $$\int \frac{3 u}{2 u+7} d u$$. Answer: The indefinite integral of the given rational function is $$\frac{3}{4} \ln|2u+7| + C$$, where C is the constant of integration.

Step 1: Identify a similar function in the table of integrals

First, let's rewrite the given integral: $$\int \frac{3 u}{2 u+7} d u$$ Notice that it can be represented as: $$\int \frac{3}{2} \cdot \frac{u}{2 u+7} d u$$ In a table of integrals, we will most likely find an entry that has a similar form to the integral: $$\int \frac{du}{ax+b}$$ Where $$a$$ and $$b$$ are constants. In our given integral, $$a = 2$$ and $$b = 7$$.

Step 2: Use the table of integrals to find the solution

From the table of integrals, we can see that the integral for $$\int \frac{du}{ax+b}$$ is given by: $$\frac{1}{a} \ln|ax+b| + C$$ In our integral, we have an additional constant $$\frac{3}{2}$$ multiplying the function inside the integral. We can take this constant outside the integral: $$\frac{3}{2} \int \frac{du}{2u+7}$$ Now we can apply the formula from the table of integrals to our function: $$\frac{3}{2} \cdot \frac{1}{2} \ln|2u+7| + C$$

Step 3: Simplify the solution

Now we simplify our result: $$\frac{3}{4} \ln|2u+7| + C$$ So, the indefinite integral of $$\int \frac{3 u}{2 u+7} d u$$ is: $$\frac{3}{4} \ln|2u+7| + C$$

Key Concepts

Table of Integrals

In calculus, the table of integrals is an essential tool for finding indefinite integrals, which are antiderivatives of functions. Unlike derivatives, which have straightforward rules for computations, integrals often require identifying patterns and associating them with known integrals.

For instance, when faced with an integral such as \(\int \frac{3 u}{2 u+7} d u\), students can consult a table of integrals to find a formula that matches a similar form. In this case, it closely resembles \(\int \frac{du}{ax+b}\), which is a standard format with corresponding solutions in most tables.

Using a table can expedite the integration process, as it avoids the need for long, intricate calculations. However, the successful application of these tables requires a good understanding of how to manipulate expressions to match the forms provided in the table and the skill to recognize these forms quickly. This is a valuable strategy for working with indefinite integrals.

Integration Techniques

While a table of integrals is a helpful reference, there are numerous integration techniques that one must learn to effectively solve indefinite integrals. These techniques include substitution, integration by parts, partial fractions, trigonometric substitution, and others.

In the exercise \(\int \frac{3 u}{2 u+7} d u\), a simple manipulation of the integrand preceded the use of the table. By factoring out constants and rearranging terms, the integrand transformed into a form found in the table. Recognizing that \(\frac{3}{2}\) is a constant that can be taken out of the integral is a basic example of these techniques in action.

For more complex functions, these integration techniques become indispensable. Often, they are used in conjunction with tables of integrals to arrive at a solution. Having a strong understanding of these techniques ensures that students are prepared to handle a wide range of integrals.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is a fundamental operation in calculus, particularly in the context of integration. It's defined as the inverse of the exponential function, specifically, the time required to reach a certain level of continuous growth. It's written as \(\ln(x)\) for \(x > 0\).

In our exercise, the natural logarithm appears as the result of the integral \(\int \frac{du}{ax+b}\). The result, given by the table of integrals, is \(\frac{1}{a} \ln|ax+b| + C\). Upon finding the indefinite integral \(\int \frac{3 u}{2 u+7} d u\), we encounter \(\frac{3}{4} \ln|2u+7| + C\) as the final answer.

Understanding the natural logarithm is essential for integrating functions that resemble inverse growth or decay. As seen in the solution, it's crucial to apply the properties of logarithms correctly when simplifying the result of an integral that produces a natural logarithmic function.