To find the integral of \( e^x (2 + e^x)^9 \), we can use the substitution method.

1. **Substitution**:  
   Let \( u = 2 + e^x \).  
   Then, \( du = e^x \, dx \).

2. **Rewrite the Integral**:  
   Substitute \( e^x \, dx \) with \( du \) and \( 2 + e^x \) with \( u \):
   \[
   \int e^x (2 + e^x)^9 \, dx = \int u^9 \, du
   \]

3. **Integrate**:  
   Now integrate \( u^9 \):
   \[
   \int u^9 \, du = \frac{u^{10}}{10} + C
   \]

4. **Back-Substitute**:  
   Replace \( u \) with \( 2 + e^x \):
   \[
   \frac{(2 + e^x)^{10}}{10} + C
   \]

So, the integral of \( e^x (2 + e^x)^9 \, dx \) is:
\[
\int e^x (2 + e^x)^9 \, dx = \frac{(2 + e^x)^{10}}{10} + C
\]

Would you like more details on this, or have any questions?

Here are 5 related questions to explore further:

1. How do you integrate \( e^x \cdot (3 + e^x)^5 \, dx \)?
2. What is the derivative of \( \frac{(2 + e^x)^{10}}{10} \)?
3. Can you find the integral of \( e^x (4 + e^x)^7 \, dx \) using the same method?
4. How would you solve \( \int e^{2x} (1 + e^{2x})^8 \, dx \)?
5. What substitution would you use for \( \int e^{3x} (5 + e^{3x})^6 \, dx \)?

**Tip**: For integrals involving expressions like \( e^x(2+e^x)^n \), substitution often simplifies the integral significantly.

Learn how to solve the integral of e^x(2+e^x)^9 using the substitution method. This problem explores advanced integration techniques and substitution, suitable for students in Grades 11-12 or early college calculus courses.

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