Question

evaluate the integral.
\int \frac{5e^{5y}}{9 + e^{5y}} \\, dy

\int \frac{5e^{5y}}{9 + e^{5y}} \\, dy = \square

Explanation:

Step1: Use substitution method

Let \( u = 9 + e^{5y} \). Then, find the derivative of \( u \) with respect to \( y \).
The derivative of \( u \) with respect to \( y \) is \( \frac{du}{dy}=5e^{5y} \), so \( du = 5e^{5y}dy \).

Step2: Substitute into the integral

The integral \( \int\frac{5e^{5y}}{9 + e^{5y}}dy \) can be rewritten using the substitution \( u = 9 + e^{5y} \) and \( du = 5e^{5y}dy \) as \( \int\frac{du}{u} \).

Step3: Integrate \( \int\frac{du}{u} \)

The integral of \( \frac{1}{u} \) with respect to \( u \) is \( \ln|u|+C \), where \( C \) is the constant of integration.

Step4: Substitute back \( u = 9 + e^{5y} \)

Since \( u = 9 + e^{5y} \) is always positive (because exponential functions are positive and we add 9), we can write the result as \( \ln(9 + e^{5y})+C \).

Answer:

\( \ln(9 + e^{5y}) + C \)