The given integral is:

$\int \frac{7}{x^2 + 9} \, dx$

Solution:

This integral can be solved using the standard formula for integrals of the form:

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

Here, $a^2 = 9$, so $a = 3$.

Step-by-step process:

1. Factor out the constant 7: $\int \frac{7}{x^2 + 9} \, dx = 7 \int \frac{1}{x^2 + 3^2} \, dx$

2. Apply the formula: $\int \frac{1}{x^2 + 3^2} \, dx = \frac{1}{3} \arctan\left(\frac{x}{3}\right) + C$

3. Multiply by 7: $7 \int \frac{1}{x^2 + 3^2} \, dx = \frac{7}{3} \arctan\left(\frac{x}{3}\right) + C$

Final Answer:

$\int \frac{7}{x^2 + 9} \, dx = \frac{7}{3} \arctan\left(\frac{x}{3}\right) + C$

Let me know if you have questions or would like further clarifications.

Related Questions:

1. How would the solution change if the numerator were different, say $5$ instead of $7$?
2. What if the denominator were $x^2 - 9$? How would the integral be solved?
3. How does the arctangent function arise in integration?
4. Can this integral be solved using substitution methods? If yes, how?
5. How does this solution relate to the integral of $\frac{1}{x^2 + 1}$?

Tip:

When you encounter integrals involving $x^2 + a^2$ in the denominator, always consider the arctangent formula for a quick solution!