I am unsure if I am doing the next to the last step correctly.
integral e^5x dx
g(u) = 1/6*u^6
u = 5x
du = 5
1/6 * 1/du * u^6
1/30 (5x)^6
Your result is wrong. You have used the wrong formula.
Notice that e^(5x) is NOT a power function. It is an exponential function.
The correct answer is e^(5x)/5+C.