Problem 79 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(71-84\) $$ \int \frac{x+4}{x^{2}+4} d x $$

Step-by-Step Solution

Answer

\(\int \frac{x+4}{x^{2}+4} \, dx = \frac{1}{2}\ln|x^2+4| + 2\arctan\left(\frac{x}{2}\right) + C\).

Step 1: Identify the Form of the Integral

The integral is of the form \(\int \frac{f'(x)}{f(x)} \, dx\), where \(f(x) = x^2 + 4\) and \(f'(x) = 2x\). This suggests using a substitution method.

Step 2: Perform Suitable Substitution

Let \(u = x^2 + 4\), thus \(\frac{du}{dx} = 2x\) or \(du = 2x \, dx\). In our integral, we have \(x \, dx\), so rewrite the integral to include our substitution properties.

Step 3: Adjust the Integrand for Substitution

Rewrite the integral as: \(\int \frac{x}{x^2 + 4} \, dx + \int \frac{4}{x^2 + 4} \, dx\). Now focus on \(\int \frac{x}{x^2 + 4} \, dx\) and use substitution.

Step 4: Apply Substitution to First Part

Use \(u = x^2 + 4\) and \(x \, dx = \frac{1}{2} du\). Substitute into the integral: \(\int \frac{x}{u} \, dx = \frac{1}{2} \int \frac{1}{u} \, du\). This simplifies to \(\frac{1}{2} \ln |u| + C_1 = \frac{1}{2} \ln |x^2 + 4| + C_1\).

Step 5: Evaluate the Second Part

Now consider \(\int \frac{4}{x^2 + 4} \, dx\). Rewrite as \(\int \frac{4}{x^2 + (2)^2} \, dx\). This fits the formula \(\int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a}\arctan\left(\frac{x}{a}\right) + C\).

Step 6: Integrate Using Arctangent Formula

Apply the formula: \(\int \frac{4}{x^2 + 4} \, dx = 4 \times \frac{1}{2} \arctan\left(\frac{x}{2}\right) = 2 \arctan\left(\frac{x}{2}\right) + C_2\).

Step 7: Combine Results

Now combine the results of both integrals: \(\frac{1}{2} \ln |x^2 + 4| + 2 \arctan\left(\frac{x}{2}\right) + C\), where \(C = C_1 + C_2\).

Key Concepts

Integrals

Integrals are one of the fundamental concepts in calculus. They are used to find areas under curves, among many other applications.
An integral is essentially the reverse operation of differentiation.
In simpler terms, it takes you from a derivative back to the original function, or something close to it. To solve an integral, we often look for patterns or rules that can simplify the process. This is where techniques such as substitution come into play. Techniques are tools in our calculus toolkit to tackle different kinds of integrals successfully:

• Definite integrals calculate the area under a curve within specific limits, from one point to another.
• Indefinite integrals, like the one we deal with here, are concerned with finding the general form of an antiderivative.

Through this process, constants of integration are included since there are infinitely many functions with the same derivative, distinguished only by these constants.

Substitution Method

The substitution method is a clever technique for solving integrals, especially when the integrand (the function being integrated) isn't in an immediately recognizable form. It simplifies the integration process by transforming the integral into a simpler or standard form.
Consider a common situation where it is possible to identify a part of the integrand as a derivative of another function. This opens up the opportunity to make a substitution, usually represented by a new variable such as \( u \).
Here’s the general process:

• Identify a substitution that simplifies the function. In this case, we let \( u = x^2 + 4 \).
• Determine the differential of \( u \), or \( du \). For our example, \( du = 2x \, dx \).
• Rewrite the entire integral in terms of \( u \) to make the integration straightforward.

This method not only aids in simplifying complex expressions but also helps apply other integration formulas more easily, like the arctangent formula seen in the exercise.

Arctangent Formula

The arctangent formula is crucial for handling integrals with specific types of expressions, typically in the form \( \int \frac{1}{x^2 + a^2} \, dx \). This integral is related to the inverse trigonometric functions, specifically the arctan or arctangent.The formula is:\[\int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C\]Where \( a \) is a constant and \( C \) is the integration constant. It's a direct formula that gives us the integral of a function that might otherwise be tricky to evaluate directly.
In the exercise, this formula is applied to evaluate the part of the integral \( \int \frac{4}{x^2 + 4} \, dx \). Recognizing this form allows easy application of the arctangent formula, giving a straightforward solution for a potentially complex problem.