determine the integral by making the appropriate substitution.\n int \frac { e ^ { 5 x } } { 3 + 7 e ^ { 5 x…

determine the integral by making the appropriate substitution.\n int \frac { e ^ { 5 x } } { 3 + 7 e ^ { 5 x } } d x \n(use parentheses to clearly denote the argument of each function.)\n int \frac { e ^ { 5 x } } { 3 + 7 e ^ { 5 x } } d x =

determine the integral by making the appropriate substitution.\n int \frac { e ^ { 5 x } } { 3 + 7 e ^ { 5 x } } d x \n(use parentheses to clearly denote the argument of each function.)\n int \frac { e ^ { 5 x } } { 3 + 7 e ^ { 5 x } } d x =

Answer

Explanation:

Step1: Split the integral

$$\int\frac{e^{5x}}{3 + 7e^{5x}}dx=\frac{1}{5}\int\frac{5e^{5x}}{3 + 7e^{5x}}dx$$ Let (u = 3+7e^{5x}), then (du=35e^{5x}dx), and (e^{5x}dx=\frac{du}{35})

Step2: Substitute and integrate

$$\frac{1}{5}\int\frac{5e^{5x}}{3 + 7e^{5x}}dx=\frac{1}{5}\times\frac{1}{7}\int\frac{du}{u}$$ Using the formula (\int\frac{1}{u}du=\ln|u|+C) $$\frac{1}{35}\ln|u|+C$$

Step3: Substitute back

Substitute (u = 3 + 7e^{5x}) back into the expression $$\frac{1}{35}\ln\left(3 + 7e^{5x}\right)+C$$

Answer:

(\frac{1}{35}\ln\left(3 + 7e^{5x}\right)+C)