The integral given in the image is:

$\int x(x-2)^5 \, dx$

Solution:

To solve this, we can use the substitution method.

1. Let $u = x - 2$.
   Then, $du = dx$, and $x = u + 2$.

2. Substitute into the integral:

   $\int x (x-2)^5 \, dx = \int (u+2)u^5 \, du$

3. Expand $(u+2)u^5$:

   $(u+2)u^5 = u^6 + 2u^5$

4. Now the integral becomes:

   $\int u^6 \, du + 2 \int u^5 \, du$

5. Integrate term by term:

   $\int u^6 \, du = \frac{u^7}{7}, \quad \int u^5 \, du = \frac{u^6}{6}$

   So:

   $\int u^6 \, du + 2 \int u^5 \, du = \frac{u^7}{7} + \frac{2u^6}{6}$

   Simplify:

   $= \frac{u^7}{7} + \frac{u^6}{3}$

6. Substitute back $u = x-2$:

   $\frac{u^7}{7} + \frac{u^6}{3} = \frac{(x-2)^7}{7} + \frac{(x-2)^6}{3}$

7. Add the constant of integration $C$:

   $\int x(x-2)^5 \, dx = \frac{(x-2)^7}{7} + \frac{(x-2)^6}{3} + C$

Final Answer:

$\boxed{\int x(x-2)^5 \, dx = \frac{(x-2)^7}{7} + \frac{(x-2)^6}{3} + C}$

Would you like me to explain any specific part in more detail or verify the solution step-by-step?

Related Questions:

1. How does substitution help simplify integrals like this?
2. Can this integral be solved without substitution? If so, how?
3. What is the significance of the constant of integration $C$?
4. How do you check if the final solution is correct?
5. What are other methods to approach this type of polynomial integral?

Tip:

Always double-check the substitution step and back-substitute carefully to avoid missing terms!