Mathematics

This integral has the form $\frac{1}{u^2 + a^2}$, which is a standard arctangent pattern. The key is recognizing that the denominator is already a sum of squares—just not in the most obvious form—so you can use the substitution $u = x - 2$ to transform it into the classic $\int \frac{du}{u^2 + a^2} = \frac{1}{a}\arctan\left(\frac{u}{a}\right) + C$ formula. Once you identify $a = 3$, the rest follows directly.

This integral has the form $\frac{1}{u^2 + a^2}$, which is a standard arctangent pattern. The key is recognizing that the denominator is already a sum of squares—just not in the most obvious form—so you can use the substitution $u = x - 2$ to transform it into the classic $\int \frac{du}{u^2 + a^2} = \frac{1}{a}\arctan\left(\frac{u}{a}\right) + C$ formula. Once you identify $a = 3$, the rest follows directly.