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The integral provided is:

\[
\int \frac{x^7 e^x - 3x^6}{x^7} \, dx
\]

To simplify this, we can first break down the expression inside the integral:

\[
\frac{x^7 e^x - 3x^6}{x^7} = e^x - \frac{3x^6}{x^7} = e^x - \frac{3}{x}
\]

So, the integral becomes:

\[
\int \left( e^x - \frac{3}{x} \right) dx
\]

Now we can integrate each term separately:

1. **Integral of \( e^x \):**
   \[
   \int e^x \, dx = e^x
   \]

2. **Integral of \( -\frac{3}{x} \):**
   \[
   \int -\frac{3}{x} \, dx = -3 \ln |x|
   \]

Putting these results together:

\[
\int \left( e^x - \frac{3}{x} \right) dx = e^x - 3 \ln |x| + C
\]

where \( C \) is the constant of integration.

### Final Answer:
\[
\int \frac{x^7 e^x - 3x^6}{x^7} \, dx = e^x - 3 \ln |x| + C
\]

Would you like more details on any step or have further questions?

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Here are 5 related questions to expand understanding:

1. What is the process for breaking down complex fractions before integration?
2. How can we confirm that the integration steps taken are correct?
3. What happens if there are different exponents in the terms within an integral?
4. How would you approach integrating functions involving trigonometric or logarithmic components?
5. What are common integration techniques for products of functions?

**Tip:** Simplifying the integrand first can make the integration process much easier, especially when dealing with fractions.

Evaluate the following indefinite integral: ∫ (x^7 * e^x - 3x^6) / x^7 dx

Learn how to evaluate the indefinite integral ∫ (x^7 e^x - 3x^6) / x^7 dx with step-by-step simplification and integration techniques. This calculus problem explores integration of exponential and rational functions, suitable for college-level calculus.

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