\( \int \frac { d x } { 4 + x ^ { 2 } } d x | \)

To solve the integral \( \int \frac{dx}{4 + x^2} \), we can use a trigonometric substitution or recognize this as a standard integral. The integral can be simplified by rewriting it as follows: Using the formula \( \int \frac{dx}{a^2 + x^2} = \frac{1}{a} \tan^{-1} \left( \frac{x}{a} \right) + C \), where \( a^2 = 4 \) (thus \( a = 2 \)), we find: \[ \int \frac{dx}{4 + x^2} = \frac{1}{2} \tan^{-1} \left( \frac{x}{2} \right) + C. \] So, the final result is: \[ \frac{1}{2} \tan^{-1} \left( \frac{x}{2} \right) + C. \] This technique of recognizing standard forms can make solving integrals like this one a whole lot smoother and faster! It's like having a cheat sheet in math class. And, if you dive into polar coordinates, this integral reappears in interesting contexts like calculating areas of circular sectors or even in wave functions in physics. Mathematics is connected in so many delightful ways!