91Ó°ÊÓ

Logarithmic integration is a method used to evaluate integrals that involve logarithmic functions. In many calculus problems, integrals feature expressions like \( \log_{b} x \). These types of integrals can sometimes be simplified further using substitutions, allowing for easier integration.
For example, when faced with \( \int \frac{1}{x(\log_{8} x)^2} \, dx \), we identify the presence of the logarithmic expression \( \log_{8} x \). By setting this portion equal to a new variable, say \( u \), we can streamline the process. Later, through integration by substitution, we change the variables and simplify the integral into a more manageable form. This highlights how identifying a logarithmic part can lead to a simpler integration path.

U-substitution is a powerful technique in integration, often allowing us to transform a complex integral into a simpler form. This process involves substituting a portion of the integral with a single variable, typically \( u \).
For the given exercise, we start by defining \( u = \log_{8} x \). This substitution choice is crucial because it helps simplify the integral where \( du \) corresponds to the derivative of \( \log_{8} x \), which is \( \frac{1}{x \ln 8} \, dx \).
Thus, we find that:

• \( du = \frac{1}{x \ln 8} \, dx \)
• \( dx = x \ln 8 \, du \)

Using these expressions, the original integral can be rewritten with \( u \) to make the integration process easier. Eventually, after solving, we switch back to the original variable to provide the final answer.

The power rule is a fundamental aspect of integration. This rule is applicable for integrating functions of the form \( x^n \). It states that \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \] provided \( n eq -1 \).
In our exercise, after substitution, the integral \( \ln 8 \int u^{-2} \, du \) arises, which fits the power rule. We substitute and simplify using the rule:
- Integrate \( u^{-2} \), which gives \( -u^{-1} \).
- Hence, the integral becomes \( \ln 8 \cdot (-u^{-1}) \). This step involves treating \( \ln 8 \) as a constant multiplier and solving the simple antiderivative. Applying the power rule turns what appears complex into a straightforward process.

Integration by substitution is a method that can effectively tackle integrals resembling a chain rule derivative. It's particularly useful for integrals where a direct application of basic rules isn't apparent.
In our example, substituting \( u = \log_{8} x \) simplifies the integral into a form that can be managed with the power rule. This strategy, known as "u-substitution," mirrors techniques in differential calculus but for antiderivatives.
Consider these steps during our substitution:

• Identify terms in the integral that seem to call for differentiation (e.g., \( \log_{8} x \)).
• Substitute these terms with a simple variable \( u \) and differentiate to find \( du \).

Completing the integration and back-substituting gives our final solution, making substitution a versatile skill in integral calculus.