To integrate the function $\frac{1}{x^2 - 10x + 25}$ with respect to $x$ using partial fractions, we can follow these steps:

Step 1: Simplify the Denominator

First, let's simplify the denominator $x^2 - 10x + 25$.

Notice that: $x^2 - 10x + 25 = (x - 5)^2$

So, the integral simplifies to: $\int \frac{1}{(x - 5)^2} \, dx$

Step 2: Integrate the Simplified Function

The function $\frac{1}{(x - 5)^2}$ can be integrated directly. We recognize that: $\frac{d}{dx} \left(-\frac{1}{x - 5}\right) = \frac{1}{(x - 5)^2}$

Thus, the integral becomes: $\int \frac{1}{(x - 5)^2} \, dx = -\frac{1}{x - 5} + C$

where $C$ is the constant of integration.

Final Answer

$\int \frac{1}{x^2 - 10x + 25} \, dx = -\frac{1}{x - 5} + C$

Would you like further details or have any questions?

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Here are 8 related questions:

1. How do you integrate more complex quadratic denominators?
2. What are partial fractions used for in integration?
3. Can you integrate a function with a cubic denominator using partial fractions?
4. How does completing the square help in integration?
5. What is the difference between proper and improper fractions in partial fractions?
6. How do you integrate $\frac{1}{(x - 2)(x - 3)}$?
7. What methods exist for integrating rational functions?
8. How do you handle repeated linear factors in partial fractions?

Tip: Always check if the denominator can be factored or simplified before applying partial fraction decomposition.