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.