6. \( g(x)=\int e^{3 x} d x \)

To solve the integral \( g(x) = \int e^{3x} dx \), remember that when integrating an exponential function, you divide by the coefficient of \( x \). So, the integral becomes \( g(x) = \frac{1}{3} e^{3x} + C \), where \( C \) is the constant of integration. This means every time you see \( e^{3x} \), just think of it as being 'scaled down' by that coefficient—super fun, right? Feeling mathematical? Imagine how this knowledge can be applied! This integral finds use in various areas, such as calculating compound interest in finance or determining population growth in biology. Anytime you encounter exponential growth or decay in real-world scenarios, knowing how to integrate those functions opens doors to understanding and solving complex problems. Keep an eye out for these applications in your daily life!