Problem 99 - Calculus and its applications

Question

Evaluate. $$ \int \frac{d x}{x(\ln x)^{4}} $$

Step-by-Step Solution

Answer

The integral evaluates to \(-\frac{1}{3(\ln x)^3} + C\)."

Step 1: Identify the Type of Integration

The integral \( \int \frac{d x}{x(\ln x)^{4}} \) is a non-standard integral, which indicates that a straightforward substitution or integration by parts might be necessary given the functions involved.

Step 2: Choose an Appropriate Substitution

We identify \( \ln x \) as a potential substitution because the function involves terms \( \frac{1}{x(\ln x)^4} \). Let \( u = \ln x \), which implies \( du = \frac{1}{x}dx \). This substitution simplifies the integral.

Step 3: Change the Variable in the Integral

Substitute \( u \) and \( du \) into the integral: \[\int \frac{d x}{x(\ln x)^{4}} = \int \frac{1}{u^{4}} du \] This simplifies to integrating a simple power of \( u \).

Step 4: Integrate the New Function

The integral \( \int \frac{1}{u^{4}} du \) can be rewritten as \( \int u^{-4} du \). Integrate using the power rule to get: \[\int u^{-4} du = \frac{u^{-3}}{-3} = -\frac{1}{3u^3} + C \]

Step 5: Substitute Back to Original Variable

Since \( u = \ln x \), substitute back into the integrated result: \[-\frac{1}{3(\ln x)^3} + C\] This provides the solution in terms of \( x \).

Key Concepts

Integration by Substitution

Integration by substitution is a beloved technique among calculus students because it allows them to simplify complex integrals into more manageable ones. The idea is to change the variable of integration to make the integral easier to solve. The process is similar to the reverse chain rule.
Here's how it works:

• Identify a part of the integrand (the expression under the integral sign) that can be substituted with a single variable, often called "u." This part usually involves a composition of functions.
• Calculate the differential of your chosen substitute, which involves differentiating "u" with respect to the original variable.
• Rewrite the entire integral in terms of "u" and "du."
• Integrate with respect to "u."
• Finally, substitute back the original variable to express the solution in terms of the initial variable.

In our example, by setting \( u = \ln x \), the integral dramatically simplified to \( \int \frac{1}{u^4} \ du \), allowing us to tackle it easily with the power rule.

Non-standard Integral

A non-standard integral is one that doesn't fit neatly into common integration formulas or rules like basic power integrals, trigonometric integrals, or exponential integrals. These integrals often require imaginative techniques such as integration by parts or substitution, akin to our problem where the integral \( \int \frac{d x}{x(\ln x)^{4}} \) didn't seem straightforward.
Why wasn't it standard? Because the presence of both \( x \) and \( \ln x \) didn't allow for immediate use of simple rules. This pushed us to find an innovative substitution approach. Non-standard integrals challenge your understanding and require you to analyze the integral closely to find patterns or substitutions that can simplify the problem.

Power Rule in Integration

The power rule is an essential tool in integration, particularly for integrals of the form \( \int x^n \, dx \). It's a straightforward rule:

• Increase the exponent by one.
• Divide by the new exponent.
• Add the constant of integration, \( C \).

In mathematical terms, the rule is expressed for \( n eq -1 \) as:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
The utility of this rule was seen in the original problem after substituting, transforming the integral into \( \int u^{-4} \, du \), which was a nice fit for the power rule. We integrated to get:
\[ \int u^{-4} \, du = \frac{u^{-3}}{-3} = -\frac{1}{3u^3} + C\]This highlights how even challenging integrals can often be brought back to fundamentals with substitution, leading to straightforward solutions with classic integration rules.