We are tasked with evaluating the integral:

$\int \frac{e^x}{(9 + 8e^x)^3} \, dx.$

Step-by-step solution:

1. Substitution: Let: $u = 9 + 8e^x.$ Then, the derivative of $u$ with respect to $x$ is: $\frac{du}{dx} = 8e^x \quad \text{or} \quad du = 8e^x \, dx.$

   Therefore, $e^x \, dx = \frac{1}{8} \, du$.

2. Substitute into the integral: The integral becomes: $\int \frac{e^x}{(9 + 8e^x)^3} \, dx = \int \frac{\frac{1}{8} \, du}{u^3}.$

   Factor out $\frac{1}{8}$: $\frac{1}{8} \int u^{-3} \, du.$

3. Evaluate the integral: The integral of $u^{-3}$ is: $\int u^{-3} \, du = \frac{u^{-2}}{-2} = -\frac{1}{2u^2}.$

   So, the result becomes: $\frac{1}{8} \cdot \left( -\frac{1}{2u^2} \right) = -\frac{1}{16u^2}.$

4. Back-substitute $u = 9 + 8e^x$: Replace $u$ with $9 + 8e^x$: $-\frac{1}{16u^2} = -\frac{1}{16(9 + 8e^x)^2}.$

5. Add the constant of integration: The final answer is: $-\frac{1}{16(9 + 8e^x)^2} + C.$

Final Answer:

$\boxed{-\frac{1}{16(9 + 8e^x)^2} + C.}$

Would you like a detailed explanation for any of these steps?

Related Questions:

1. How does substitution simplify solving integrals like this?
2. What happens if we substitute $u = e^x$ instead?
3. Why does the integral of $u^{-3}$ lead to a power rule formula?
4. How do you verify the solution of an integral by differentiation?
5. What are other examples of integrals that involve exponential expressions?

Tip:

Always double-check substitutions to ensure the variable transformation and differential match perfectly!