Evaluate the integral: I = integral dx/(x² + 9)

Evaluate the integral: $$ I = \int \frac{dx}{x^2 + 9}. $$

Explanation of Method

The denominator is already in the standard form $x^2+a^2$ with $a^2=9$, so no algebraic manipulation is needed before applying the known formula $\int \frac{dx}{x^2+a^2}=\frac1a\arctan\left(\frac{x}{a}\right)$.

Recognize the denominator as already matching the standard form $x^2+a^2$.
Identify $a$ from $a^2$.
Apply the standard arctangent integral formula directly.

Step 1 · Recognize the standard form

We recognize that this integral has the standard form:

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

In our case, $a^2 = 9$, so $a = 3$.

Step 2 · Apply the standard result

Using the standard result for the integral, we get:

$$ I = \frac{1}{3}\arctan\left(\frac{x}{3}\right) + C, $$

where $C$ is the constant of integration.

Summary Table

Quantity	Value
$a^2$	$9$
$a$	$3$
Antiderivative	$\tfrac13\arctan(x/3)$

Additional Notes

The integrand is defined for all real $x$, since $x^2+9 \geq 9 > 0$ always.
This is the most direct case of the arctangent formula, with no algebraic setup required — compare with Problem 38 and Problem 40, where a coefficient or irrational $a$ first needs to be extracted.
The result can be checked by differentiating $\tfrac13\arctan(x/3)$ with the chain rule and confirming it returns $\dfrac{1}{x^2+9}$.

Result

The solution to the integral is:

$$ I = \frac{1}{3}\arctan\left(\frac{x}{3}\right) + C. $$