$\int e^{8 x} d x$

Step 1 ：Given the integral \(\int e^{8x} dx\)

Step 2 ：We use the rule of integration for exponential functions, which states that the integral of \(e^{ax} dx\) is \(\frac{1}{a}e^{ax} + C\), where \(C\) is the constant of integration.

Step 3 ：So, for the integral of \(e^{8x} dx\), \(a = 8\).

Step 4 ：Therefore, \(\int e^{8x} dx = \frac{1}{8}e^{8x} + C\)

Step 5 ：To check if this result is correct, we differentiate it and see if we get back the original function \(e^{8x}\).

Step 6 ：The derivative of \(\frac{1}{8}e^{8x} + C\) is \(e^{8x}\), which is indeed the original function.

Step 7 ：Therefore, the result is correct and the integral of \(e^{8x} dx\) is \(\boxed{\frac{1}{8}e^{8x} + C}\)