does \int ((x-1)(x-3))/((x-1)(x-3)(x-4))dx converge or diverge

Let's analyze the integral:

\[
\int \frac{(x-1)(x-3)}{(x-1)(x-3)(x-4)} \, dx
\]

Simplify the integrand:

\[
\frac{(x-1)(x-3)}{(x-1)(x-3)(x-4)} = \frac{1}{x-4}
\]

So, the integral becomes:

\[
\int \frac{1}{x-4} \, dx
\]

This is a standard logarithmic integral, and its solution is:

\[
\ln |x-4| + C
\]

This function does not have any problems in terms of convergence in the general case, as long as the limits of integration do not include \(x = 4\), where the function has a vertical asymptote.

### Conclusion:
- The integral converges if the limits of integration do not include \(x = 4\) (since the function has a vertical asymptote at \(x = 4\)).
- The integral diverges if the limits of integration approach or include \(x = 4\).

Would you like details on improper integrals or have any questions about this?

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Here are 5 related questions to further your understanding:
1. How does the behavior of integrals change near vertical asymptotes?
2. What is the difference between a converging and diverging integral?
3. How do we handle improper integrals with discontinuities?
4. What are the conditions for a logarithmic function to cause divergence?
5. How does the choice of limits of integration affect the outcome of this integral?

**Tip:** Always check for discontinuities or singularities in the integrand before evaluating the integral over a given range.

does ∫ ((x-1)(x-3))/((x-1)(x-3)(x-4)) dx converge or diverge

Analyze whether the integral ∫ ((x-1)(x-3))/((x-1)(x-3)(x-4)) dx converges or diverges. Simplify the integrand and understand the behavior near vertical asymptotes, focusing on the logarithmic function ln |x-4| + C. Suitable for college-level calculus students.

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