The given integral is:

$\int \frac{1}{x^2 + 25} \, dx$

To solve this integral, notice that the denominator $x^2 + 25$ is a sum of squares. The standard formula for this type of integral is:

$\int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C$

Here, $a^2 = 25$, so $a = 5$. Applying the formula:

$\int \frac{1}{x^2 + 25} \, dx = \frac{1}{5} \arctan\left(\frac{x}{5}\right) + C$

Final Answer:

$\frac{1}{5} \arctan\left(\frac{x}{5}\right) + C$

Would you like further details about the derivation of this formula or related examples?

Here are 5 related questions to expand your understanding:

1. How does the formula for the integral of $\frac{1}{x^2 + a^2}$ derive from trigonometric substitution?
2. What is the difference between solving integrals involving sums vs. differences of squares?
3. Can this method be extended to complex numbers?
4. What would the integral look like if the numerator was $x$ instead of 1?
5. How does the arc tangent function relate to the unit circle?

Tip: Memorizing the standard forms of integrals like $\int \frac{1}{x^2 + a^2} dx$ is essential for solving integrals efficiently!