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To tackle the integral \( \int \frac{(\ln x)^{4}}{x} \, dx \), the method known as integration by substitution is a helpful technique. It is especially useful when the integrand (the function you are integrating) has a composite structure—meaning one function is inside another.

The process typically involves these steps:

• Choose a substitution: You replace a part of the integrand with a new variable, often denoted as \( u \). In this problem, \( u = \ln x \) simplifies our computations since its derivative, \( \frac{du}{dx} = \frac{1}{x} \), appears in the integrand.
• Express \( dx \) in terms of \( du \): Rearrange the derivative to get \( du = \frac{1}{x} \, dx \), allowing us to substitute \( dx \) in our integral.
• Transform the integral: Substitute \( u \) into the integral, converting it to \( \int u^4 \, du \).

By performing these steps, you simplify the original integral into a form that is easier to solve. After integrating, remember to resubstitute the original variable back, which for our case, turns \( u \) back into \( \ln x \). Substitution allows us to handle integrals that look difficult at first glance by recognizing patterns and performing an algebraic transformation.

The natural logarithm, denoted as \( \ln x \), is a special mathematical function crucial in calculus and mathematical analysis. It is the inverse of the exponential function, specifically the base \( e \) exponentiation.

Some core properties of \( \ln x \):

• Derivatives and Integrals: The derivative of \( \ln x \) is \( \frac{1}{x} \). Understanding this derivative is key when applying the substitution \( u = \ln x \), because it modeled the transformation \( du = \frac{1}{x} \, dx \).
• Domain: \( \ln x \) is only defined for \( x > 0 \), as logarithms of non-positive numbers aren't real.
• Growth Rate: \( \ln x \) increases slower than powers of \( x \) but faster than roots of \( x \).

The substitution of \( \ln x \) in an integral often simplifies functions that include logarithms raised to powers or multiplied by other functions, making problems more tractable by leveraging the simple derivative of \( \ln x \). This enhances our ability to integrate complex expressions through manageable substitution steps.

The power rule for integration is fundamental for solving integrals involving terms raised to a power. It states that:\[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad \text{where}\; n eq -1.\]

In integrals, this rule is invaluable when you have expressions like \( u^4 \). In the given exercise, after substituting \( u = \ln x \), the integral simplifies to \( \int u^4 \, du \).

Key points about using the power rule:

• Applicability: It is straightforward as long as the exponent isn't \(-1\), where a logarithmic integration approach would be needed instead.
• Efficiency: Significantly simplifies the integration process when direct exponentiation applies.
• Generalization: For positive and negative powers, this rule can be applied, making it versatile for a broad range of problems.

Remember, after integrating using the power rule, you'll often have to re-introduce the original variable. For instance, here, after finding \( \frac{u^5}{5} + C \), substitute back \( u \) with \( \ln x \), concluding the process with \( \frac{(\ln x)^5}{5} + C \). Reiteration of the original variable is critical for ensuring the solution addresses the variable with respect to which integration started.