Let us solve the sixth integral:

$\int (e^x + 1)^3 \cdot e^x \, dx$

Step 1: Substitution

Let: $u = e^x + 1 \implies du = e^x \, dx$

This transforms the integral into: $\int u^3 \, du$

Step 2: Evaluate the new integral

The integral of $u^3$ is: $\int u^3 \, du = \frac{u^4}{4} + C$

Step 3: Back-substitute $u$

Substitute back $u = e^x + 1$ to get: $\frac{(e^x + 1)^4}{4} + C$

Final Answer:

$\int (e^x + 1)^3 \cdot e^x \, dx = \frac{(e^x + 1)^4}{4} + C$

Let me know if you need explanations or help with another question. Here are related problems:

1. How to handle substitution for other exponential integrals?
2. How to deal with integrals where substitution doesn’t directly work?
3. What is the role of exponential growth in calculus applications?
4. How to simplify polynomial expressions raised to powers in integrals?
5. How does substitution simplify more complex integrals?

Tip: Always check if substitution reduces the integral into a polynomial—it saves time and avoids mistakes!