Question

Evaluate the following integrals. Include absolute values only when needed. $$\int \frac{\sin (\ln x)}{4 x} d x$$

Short answer

Question: Evaluate the following integral: \( \int \frac{\sin (\ln x)}{4 x} dx \) Answer: \( -\frac{1}{4} \cos(\ln x) + C \)

Step-by-step solution

Choose a substitution

Let's choose the substitution \(u = \ln x\). This means that \(e^u = x\).

Differentiate the substitution variable

Now, we need to find the differential of \(u\), which is \(du = \frac{1}{x} dx\).

Rewrite the integral in terms of \(u\)

We can rewrite the integral as $$ \int \frac{\sin (\ln x)}{4 x} d x = \int \frac{\sin (u)}{4 e^u} e^u du = \frac{1}{4} \int \sin (u) du.$$

Evaluate the integral in terms of \(u\)

Now, we can integrate the simplified integral with respect to \(u\): $$ \frac{1}{4} \int \sin (u) du = -\frac{1}{4} \cos(u) + C.$$

Replace \(u\) with the original variable

Finally, we substitute back in for \(u\) to obtain the integral in terms of \(x\): $$ -\frac{1}{4} \cos(u) + C = -\frac{1}{4} \cos(\ln x) + C. $$ So, the final result is: $$ \int \frac{\sin (\ln x)}{4 x} d x = -\frac{1}{4} \cos(\ln x) + C. $$

Key concepts

Substitution method

The substitution method is a powerful technique often used in calculus to simplify integrals. It involves replacing a part of the integral with a new variable to make the integration process easier. For this exercise, we used the substitution \(u = \ln x\), allowing us to change variables from \(x\) to \(u\). This change makes the integral simpler to evaluate.

To effectively use the substitution method:

• Identify a part of the integral that can be replaced with a simpler variable.
• Calculate the differential of the new variable; here, it was \(du = \frac{1}{x} dx\).
• Rewrite the integral in terms of the new variable before integrating.
• After evaluating, substitute back the original variable.

By following these steps, integrals that seem complex initially can become manageable.

Trigonometric integrals

Trigonometric integrals involve integrating functions that include trigonometric functions such as sine, cosine, or tangent. In our exercise, the presence of \(\sin(\ln x)\) makes the integral a trigonometric one after substitution.

The main idea is to use identities and techniques that simplify these trigonometric functions to solve the integral:

• Rewrite complex trigonometric expressions using known identities if needed.
• Integrate using standard formulas, such as \(\int \sin(u) \, du = -\cos(u) + C\) for this example.
• For more complicated functions, consider techniques such as integration by parts or further substitutions.

Understanding these methods allows you to tackle a variety of integrals involving trigonometric functions effectively.

Definite integrals

Although the provided exercise focuses on an indefinite integral, understanding definite integrals is essential. A definite integral evaluates the area under a curve between two specific limits, \([a, b]\). It is crucial for calculations involving exact quantities rather than general equations.

Key aspects of definite integrals include:

• Evaluating the antiderivative or indefinite integral first.
• Applying the Fundamental Theorem of Calculus, which states that \(\int_a^b f(x) \, dx = F(b) - F(a)\).
• Ensuring the integrand and the limits are correctly substituted and evaluated.
• Addressing any discontinuities within the limits.

This knowledge complements indefinite integrals by allowing you to derive specific numerical results.

Indefinite integrals

The focus of the original exercise is on indefinite integrals. These integrals represent a family of functions and contain arbitrary constants (usually \(C\)). They are essential for finding general solutions to integration problems.

With indefinite integrals, the key steps are:

• Understand the function to be integrated and apply appropriate techniques, such as substitution.
• Evaluate the antiderivative, ensuring all parts of the function are accurately handled.
• Include a constant of integration (\(C\)), reflecting the indefinite nature of the solution.
• Verify your solution, especially by differentiating to ensure you recover the original function.

Mastery of indefinite integrals sets the foundation for approaching both simple and complex calculus problems.