Evaluate the integral.

Integral from nothing to nothing e Superscript x Baseline left parenthesis e Superscript x Baseline plus 7 right parenthesis Superscript negative 3 Baseline dx∫exex+7−3 dx

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Part 1

A.

StartFraction 1 Over left parenthesis e Superscript x Baseline plus 7 right parenthesis Superscript 4 EndFraction1ex+74plus+C

B.

minus−StartFraction 2 Over left parenthesis e Superscript x Baseline plus 7 right parenthesis squared EndFraction2ex+72plus+C

C.

StartFraction 1 Over 4 left parenthesis e Superscript x Baseline plus 7 right parenthesis Superscript 4 EndFraction14ex+74plus+C

D.

minus−StartFraction 1 Over 2 left parenthesis e Superscript x Baseline plus 7 right parenthesis squared EndFraction12ex+72plus+C




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

---

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.

Evaluate the integral: ∫e^x(e^x + 7)^{-3} dx

This guide walks you through solving the integral ∫e^x(e^x + 7)^{-3} dx using substitution and the power rule. Follow the step-by-step breakdown suitable for undergraduate calculus students.

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