This is the integral problem:

$\int x(x - 1)^5 \, dx$

Solution:

We will use substitution to simplify the problem.

1. Let: $u = x - 1 \quad \Rightarrow \quad du = dx \quad \text{and} \quad x = u + 1.$

2. Substituting $x = u + 1$ into the integral: $\int x(x - 1)^5 \, dx = \int (u + 1)u^5 \, du.$

3. Expand $(u + 1)u^5$: $(u + 1)u^5 = u^6 + u^5.$

4. The integral becomes: $\int u^6 + u^5 \, du = \int u^6 \, du + \int u^5 \, du.$

5. Integrate term by term: $\int u^6 \, du = \frac{u^7}{7}, \quad \int u^5 \, du = \frac{u^6}{6}.$

6. Combine the results: $\frac{u^7}{7} + \frac{u^6}{6} + C.$

7. Substitute back $u = x - 1$: $\frac{(x-1)^7}{7} + \frac{(x-1)^6}{6} + C.$

Final Answer:

$\boxed{\frac{(x-1)^7}{7} + \frac{(x-1)^6}{6} + C}$

Would you like a step-by-step walkthrough or have any additional questions?

Related Questions:

1. How do you determine when substitution is appropriate for an integral?
2. Can this problem be solved using integration by parts instead?
3. How does substitution simplify integration in general?
4. What are other examples of similar polynomial integrals?
5. How do you verify the result of this integral?

Tip:

Always check the substitution step carefully by ensuring $du$ aligns with the transformed integral limits or variable replacements.