Problem
Evaluate the integral. \[ \int \frac{5 e^{5 t}}{1+e^{5 t}} d t \]
Solution
Step 1 ：Let \( u = 1 + e^{5t} \). Then, \( du = 5e^{5t} dt \).
Step 2 ：Rewrite the integral in terms of u: \( \int \frac{5 e^{5 t}}{1+e^{5 t}} d t = \int \frac{1}{u} du \).
Step 3 ：Integrate to get: \( \int \frac{1}{u} du = ln|u| + C \).
Step 4 ：Substitute back \( u = 1 + e^{5t} \) to get the final answer: \( ln|u| + C = ln|1 + e^{5t}| + C \).
Step 5 ：\(\boxed{ln|1 + e^{5t}| + C}\) is the solution to the integral.
