Problem 103 - Calculus and its applications

Question

Evaluate. $$ \int \frac{\left[(\ln x)^{2}+3(\ln x)+4\right]}{x} d x, x>0 $$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{(\ln x)^3}{3} + \frac{3(\ln x)^2}{2} + 4(\ln x) + C \).

Step 1: Identify the Integration Type

Notice that the integrand has the form \( \frac{f(\ln x)}{x} \). This suggests the use of substitution with \( u = \ln x \). Then \( du = \frac{1}{x} dx \).

Step 2: Perform the Substitution

Substitute \( u = \ln x \), then \( du = \frac{1}{x} dx \). The integrand becomes \( \left[(u)^2 + 3u + 4\right] du \). Thus, the integral is now \( \int [(u)^2 + 3u + 4] du \).

Step 3: Integrate with Respect to \( u \)

Integrate each term individually. The integral of \( u^2 \) is \( \frac{u^3}{3} \), the integral of \( 3u \) is \( \frac{3u^2}{2} \), and the integral of \( 4 \) is \( 4u \). Thus, the integration gives \( \frac{u^3}{3} + \frac{3u^2}{2} + 4u + C \), where \( C \) is the constant of integration.

Step 4: Substitute Back for \( x \)

Replace \( u \) with \( \ln x \) to express the integral in terms of \( x \). The result is \( \frac{(\ln x)^3}{3} + \frac{3(\ln x)^2}{2} + 4(\ln x) + C \).

Key Concepts

Integration Techniques

Integration techniques help us solve a variety of integrals. These methods are essential tools in integral calculus. They allow us to find antiderivatives or evaluate definite and indefinite integrals.

There are several common techniques:

• Substitution Method
• Integration by Parts
• Partial Fraction Decomposition
• Trigonometric Substitution

Choosing the right technique depends on the form of the integrand. For our given problem, we notice a pattern that pairs well with the Substitution Method.

Learning to identify these patterns with practice makes integration a more straightforward process. Each technique adds a new layer of tools to our calculus toolbox.

Substitution Method

The substitution method simplifies integrals by changing variables. When used correctly, it transforms a complicated integral into a simpler one.

The idea is to select a substitution that simplifies the integrand. In our example, we chose:

• Let \(u = \ln x\)
• Then \(du = \frac{1}{x} dx\)

This substitution rearranges the original integral into a polynomial in terms of \(u\), which is much simpler to evaluate.

The power of the substitution method lies in its ability to turn a daunting problem into one with a straightforward solution. With enough practice, identifying these types of substitutions can become second nature.

Definite Integrals

Definite integrals evaluate the area under a curve between two points. They are represented by the integral sign with limits, such as \( \int_a^b f(x) \, dx \).

While this exercise focused on an indefinite integral, definite integrals are essential for calculating exact values. They are used in numerous applications from physics to economics.

Evaluating a definite integral requires finding the antiderivative of the function, then applying the limits:

• Calculate the antiderivative: \( F(x) \)
• Apply the limits: \( F(b) - F(a) \)

Understanding definite integrals adds depth to your problem-solving skills, allowing for practical applications of calculus concepts in real-world scenarios.

Indefinite Integrals

Indefinite integrals are used to find the antiderivative of functions. They provide a family of functions that describes the original function from which the derivative was taken.

The notation \( \int f(x) \, dx \) is used for indefinite integrals. Unlike definite integrals, they don't have limits of integration indicating a range.

A solution to an indefinite integral includes a constant of integration \( C \) because antiderivatives are not unique. For our exercise, the indefinite integral is:

• \( \frac{(\ln x)^3}{3} + \frac{3(\ln x)^2}{2} + 4(\ln x) + C \)

Mastering indefinite integrals is crucial for solving various problems where the process requires reversing differentiation.