Problem 31 - College Algebra and Calculus: An Applied Approach

Question

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int \frac{\ln x}{x^{2}} d x $$

Step-by-Step Solution

Answer

The indefinite integral of the function \(\frac{\ln x}{x^{2}}\) dx is -ln(x) - x + c.

Step 1 - Identifying the formula

We will use the formula for integration by parts which is: \[ \int udv = uv - \int v du \] where u is the function we wish to integrate and dv is the function which we can easily integrate. In this case, we will set: (1) \(u = ln(x)\), therefore, \(du = \(\frac{1}{x}\)dx\) (2) \(dv = \rac{1}{x^2} dx\), therefore, \(v = -\frac{1}{x} \)

Step 2 - Applying the formula

Substituting u, du, v and dv in the integration by parts formula, we get: \[\int udv = uv - \int v du = (ln(x) * -\frac{1}{x}) - \int (-\frac{1}{x} * \frac{1}{x} dx)\] which simplifies to: \[-ln(x) + \int \frac{1}{x^2} dx\].

Step 3 - Final Simplification

We now integrate and simplify the equation fully: \[-ln(x) + \int \frac{1}{x^2} dx = -ln(x) - x + c \], where c is the constant of integration.

Key Concepts

Integration by Parts

To solve certain integrals, mathematicians often rely on a technique called Integration by Parts. It's like breaking an intricate puzzle into manageable pieces. What we're essentially doing is rearranging the problem to make it easier to solve. The formula for integration by parts is:\[ \int u \, dv = uv - \int v \, du \]This formula is derived from the product rule of differentiation, but we apply it in reverse, or rather for integration here. Here's how you might approach a problem using this technique:

• Identify the parts of the integral that will be \( u \) and \( dv \). This choice is crucial for simplifying the integration process later.
• Once \( u \) and \( dv \) are chosen, find \( du \) by differentiating \( u \), and find \( v \) by integrating \( dv \).
• Substitute \( u \), \( dv \), \( du \), and \( v \) back into the integration by parts formula.
• Solve the remaining integral and simplify your equation.

For our example, using \( u = \ln x \) and \( dv = \frac{1}{x^2} \, dx \), simplifies the process of solving the integral \( \int \frac{\ln x}{x^2} \, dx \). By carefully selecting \( u \) and \( dv \), this method turns a challenging integral into something far more manageable.

Logarithmic Functions

Logarithmic functions, represented with \( \ln(x) \) in mathematics as the natural logarithm, are special because they are the inverse operations of exponentials. When dealing with integration and differentiation, understanding the behavior of logarithmic functions becomes crucial.

• The derivative of \( \ln(x) \) is \( \frac{1}{x} \), which makes integration involving logarithmic functions particularly interesting.
• Logarithmic rules can help simplify expressions, making them easier to integrate or differentiate.

In integration, whenever you see a logarithm in the integrand, you might think of using integration techniques such as integration by parts, like in the example problem. The properties of logarithms allow us to manipulate and solve integrals that initially seem complex. It's all about breaking down and rearranging the pieces, often simplifying them into something recognizable and solvable.In our exercise, recognizing \( \ln(x) \) as a potential \( u \) for integration by parts is pivotal in restructuring the integral into an easily solvable form.

Constant of Integration

In integral calculus, the Constant of Integration is essential when discussing indefinite integrals. It's a representation of all the possible constants that can be added to a function, all of which are solutions to the derivative problem.

• When you calculate an indefinite integral, it's not just one function you're finding; it's an entire family of functions. Each differs by this constant \( c \).
• The reason we include the constant \( c \) is because differentiation eliminates constants in the process, leaving us with an unknown that must be represented in the antiderivative.

Imagine a line graph of functions, all of which are similar, just shifted up or down by various amounts. That's what the constant represents. It's important to note it at every step of solving indefinite integrals, to ensure your solution represents all possible original functions.In the solution given for the integral \( \int \frac{\ln x}{x^2} \, dx \), the constant of integration \( c \) is added at the end of the process, affirming the solution's completeness. It's a subtle but vital part of the final answer, ensuring that no potential solutions are missed.