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solve with partial fraction differentiation.

To solve the integral provided using partial fraction decomposition:

\[ \int \frac{5}{x(x^2 + 6)^2} \, dx \]

We start by decomposing the fraction into simpler parts. We express \(\frac{5}{x(x^2 + 6)^2}\) in terms of partial fractions as follows:

\[ \frac{5}{x(x^2 + 6)^2} = \frac{A}{x} + \frac{Bx + C}{x^2 + 6} + \frac{Dx + E}{(x^2 + 6)^2} \]

To find \(A\), \(B\), \(C\), \(D\), and \(E\), we multiply both sides by the denominator \(x(x^2 + 6)^2\), leading to:

\[ 5 = A(x^2 + 6)^2 + (Bx + C)x(x^2 + 6) + (Dx + E)x \]

Now, we can equate the coefficients from each side of the equation for different powers of \(x\) to solve for the constants. This process involves setting values for \(x\) and equating like terms, or comparing coefficients directly if the expression is expanded fully.

Let's start the process of finding these constants.

Solve the integral: \( \int \frac{5}{x(x^2 + 6)^2} \, dx \) using partial fraction decomposition.

This page provides a detailed step-by-step solution to the integral \( \int \frac{5}{x(x^2 + 6)^2} \, dx \) using partial fraction decomposition. The problem involves advanced calculus techniques suitable for undergraduate students, focusing on the method of integrating rational functions.

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