Question

evaluate the integral.
∫ \frac{6e^{6s}}{6 + e^{6s}} ds

∫ \frac{6e^{6s}}{6 + e^{6s}} ds = \square

Explanation:

Step1: Use substitution method

Let \( u = 6 + e^{6s} \), then find the derivative of \( u \) with respect to \( s \).
The derivative of \( u \) with respect to \( s \) is \( \frac{du}{ds}=6e^{6s} \), so \( du = 6e^{6s}ds \).

Step2: Substitute into the integral

The integral \( \int\frac{6e^{6s}}{6 + e^{6s}}ds \) can be rewritten as \( \int\frac{du}{u} \) (since \( 6e^{6s}ds = du \) and \( 6 + e^{6s}=u \)).

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

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

Step4: Substitute back \( u = 6 + e^{6s} \)

Since \( 6 + e^{6s}>0 \) for all real \( s \), we can drop the absolute value sign. So the integral becomes \( \ln(6 + e^{6s})+C \).

Answer:

\( \ln(6 + e^{6s}) + C \) (where \( C \) is the constant of integration)