To evaluate the given integral:

$\int e^x (e^x + 7)^{-3} \, dx$

Let's proceed step by step.

Step 1: Substitution

We can use the substitution method. Set: $u = e^x + 7$ Then, the derivative of $u$ with respect to $x$ is: $\frac{du}{dx} = e^x$ This implies that: $du = e^x \, dx$

Step 2: Rewrite the Integral

Substituting $u = e^x + 7$ and $du = e^x \, dx$, the integral becomes: $\int u^{-3} \, du$

Step 3: Apply the Power Rule for Integration

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

Step 4: Substitute Back $u = e^x + 7$

Now, substitute back $u = e^x + 7$: $-\frac{1}{2} (e^x + 7)^{-2}$

Step 5: Add the Constant of Integration

Don't forget to add the constant of integration $C$: $-\frac{1}{2} (e^x + 7)^{-2} + C$

Final Answer:

Thus, the solution to the integral is: $-\frac{1}{2} (e^x + 7)^{-2} + C$

This matches option D.

Answer:

D. $-\frac{1}{2} (e^x + 7)^{-2} + C$

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Do you have any questions or want more details about the steps? Here are 5 related questions to consider:

1. How does substitution work in integral calculus?
2. What is the power rule for integration and how is it applied here?
3. How do you recognize which substitution to use for an integral?
4. What is the importance of the constant of integration in indefinite integrals?
5. Can you derive this integral without substitution?

Tip: When using substitution in integrals, always remember to change all parts of the integrand, including the differential $dx$, before integrating.