Question

determine the integral by making the appropriate substitution.

$$ int \frac { e ^ { x } } { 1 + 7 e ^ { x } } d x $$

$$ int \frac { e ^ { x } } { 1 + 7 e ^ { x } } d x = $$
(use parentheses to clearly denote the argument of each function.)

Explanation:

Step1: Let \( u = 1 + 7e^{x} \)

Differentiate \( u \) with respect to \( x \): \( du=7e^{x}dx \), so \( e^{x}dx=\frac{1}{7}du \)

Step2: Substitute into the integral

The integral \( \int\frac{e^{x}}{1 + 7e^{x}}dx \) becomes \( \int\frac{1}{u}\cdot\frac{1}{7}du \)

Step3: Integrate

\( \frac{1}{7}\int\frac{1}{u}du=\frac{1}{7}\ln|u|+C \)

Step4: Substitute back \( u = 1 + 7e^{x} \)

\( \frac{1}{7}\ln(1 + 7e^{x})+C \)

Answer:

\( \frac{1}{7}\ln(1 + 7e^{x})+C \)