Evaluate the integral.
\[
\int \frac{5 e^{5 t}}{1+e^{5 t}} d t
\]

Answer

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Answer

\(\boxed{ln|1 + e^{5t}| + C}\) is the solution to the integral.

Steps

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.