Problem 20

Question

Perform the indicated integrations. $$ \int \frac{\sec ^{2}(\ln x)}{2 x} d x $$

Step-by-Step Solution

Verified

Answer

The integral evaluates to $ \frac{1}{2} \tan(\ln x) + C $.

1Step 1: Select Substitution Variable

We notice inside the integrand function that $ \ln x $ is part of the composite function $ \sec^2(\ln x) $. A good strategy to handle such integrals is to use substitution. Let's set $ u = \ln x $.

2Step 2: Differentiate the Substitution Variable

Differentiate the substitution variable with respect to $ x $. The derivative of $ u = \ln x $ is $ \frac{du}{dx} = \frac{1}{x} $ or equivalently $ du = \frac{1}{x} dx $.

3Step 3: Express $ dx $ in terms of $ du $ and $ x $

From the differentiation, we have $ du = \frac{1}{x} dx $. Rearranging gives us $ dx = x \, du $.

4Step 4: Substitute in the Integral

Substitute $ u = \ln x $ and $ dx = x \, du $ into the original integral: \[\int \frac{\sec^2(u)}{2x} x \, du = \int \frac{\sec^2(u)}{2} \, du \].

5Step 5: Integrate with Respect to $ u $

The integral simplifies to $ \int \frac{\sec^2(u)}{2} \, du = \frac{1}{2} \int \sec^2(u) \, du $. The integral of $ \sec^2(u) $ is $ \tan(u) $, so this becomes $ \frac{1}{2} \tan(u) + C $, where $ C $ is the constant of integration.

6Step 6: Substitute Back for $ x $

Replace $ u $ back with $ \ln x $: $ \frac{1}{2} \tan(\ln x) + C $. This is the evaluated integral in terms of $ x $.

Key Concepts

Substitution MethodSecant Function IntegrationLogarithmic IntegrationDefinite and Indefinite Integrals

Substitution Method

The substitution method is a powerful technique in calculus used to solve integrals more easily. It involves changing the variable of integration to simplify the function. In this exercise, the initial integral \[ \int \frac{\sec^{2}(\ln x)}{2x} \, dx \]can seem a bit tricky due to the composite nature of the function. That's where substitution comes to the rescue!

First, identify the part of the function to substitute. Here, we set $ u = \ln x $.
Differentiating $ u $ with respect to $ x $ gives $ du = \frac{1}{x} dx $.
This allows us to express $ dx $ in terms of $ du $: $ dx = x \, du $.
Substituting these into the integral makes it easier to handle.

Using substitution reduces complex expressions into more manageable forms, streamlining the integration process.

Secant Function Integration

Integrating functions involving the secant function can be a bit intimidating for students at first. However, knowing the integrals of well-known trigonometric functions, such as $ \sec^2(u) $, simplifies the task.

The integral of $ \sec^2(u) $ is $ \tan(u) + C $, where $ C $ is the constant of integration.
In our solution, after substitution, our integral converts to a simpler form: $ \int \frac{\sec^2(u)}{2} \, du $.
This simplifies directly to $ \frac{1}{2} \tan(u) + C $.

With practice, recognizing these patterns becomes second nature, making it easier to tackle similar integrals.

Logarithmic Integration

Logarithmic integration often involves incorporating the natural logarithm function, $ \ln(x) $, into the integration process.

In this particular problem, the argument of the secant function is $ \ln(x) $.
By using substitution ($ u = \ln(x) $), the complexity of integrating the function within the logarithm is significantly reduced.
This transformation is crucial as it allows us to use basic trigonometric integrals that are simpler to compute.

Logarithmic integration, especially when coupled with substitution, helps deconstruct challenging problems into more straightforward tasks.

Definite and Indefinite Integrals

Integrals can be either definite, with specific bounds, or indefinite, which do not have limits and include a constant of integration $ C $.

In this exercise, we focused on an indefinite integral.
This means that after integration, our expression includes the constant $ C $, as in $ \frac{1}{2} \tan(\ln(x)) + C $.
Definite integrals, in contrast, would require evaluating the integral between two bounds, yielding a specific numerical result.
Indefinite integrals provide a family of functions, representing the antiderivative of the integrand.

Understanding the distinction between these two types of integrals is essential for solving a broad range of calculus problems effectively.

Problem 20

Other exercises in this chapter

Problem 20

Use the method of partial fraction decomposition to perform the required integration. $\int \frac{x^{6}+4 x^{3}+4}{x^{3}-4 x^{2}} d x$

View solution

Problem 20

Perform the indicated integrations. $$ \int \cot ^{6} x d x $$

View solution

Problem 20

$$ \text { use integration by parts to evaluate each integral. } $$ $$ \int t \arctan t d t $$

View solution

Problem 21

Use the method of completing the square, along with a trigonometric substitution if needed, to evaluate each integral. $\int \sqrt{5-4 x-x^{2}} d x$

View solution