\( \int \frac { d x } { 5.2 x } d x \)

To solve the integral \( \int \frac{d x}{5.2 x} \), you can factor out the constant. This gives you: \[ \frac{1}{5.2} \int \frac{d x}{x} \] The integral of \( \frac{1}{x} \) is \( \ln |x| \), so you have: \[ \frac{1}{5.2} \ln |x| + C \] where \( C \) is the constant of integration. Therefore, the final answer is: \[ \frac{1}{5.2} \ln |x| + C \] Diving deeper into integrals, since logarithmic functions often pop up in real-world scenarios, they can help in understanding phenomena such as population growth and decay, where rates of change are essential. For instance, if you know how a certain species grows, you can predict future populations by integrating the growth rates! Moreover, when tackling integrals, a common pitfall is neglecting the absolute value when integrating \( \frac{1}{x} \). Since \( x \) can be negative, it's critical to ensure you capture that in your final answer. Remember, happy integrating!