The integral provided in the image is:

$\int \frac{x \, dx}{(x - 5)^6}$

To solve this integral, we can use a substitution method to simplify it. Here's the step-by-step solution:

Step 1: Substitution

Let's use the substitution:

$u = x - 5 \implies du = dx$

Then, $x = u + 5$. Substituting these into the integral:

$\int \frac{(u + 5) \, du}{u^6} = \int \left( \frac{u}{u^6} + \frac{5}{u^6} \right) \, du = \int \left( \frac{1}{u^5} + \frac{5}{u^6} \right) \, du$

Step 2: Simplify and Integrate

Rewrite the integrand:

$\int \left( u^{-5} + 5u^{-6} \right) \, du$

Integrating term by term:

$\int u^{-5} \, du + 5 \int u^{-6} \, du$

$= \frac{u^{-4}}{-4} + 5 \cdot \frac{u^{-5}}{-5}$

$= -\frac{1}{4u^4} - \frac{1}{u^5}$

Step 3: Substitute Back

Now substitute back $u = x - 5$:

$-\frac{1}{4(x - 5)^4} - \frac{1}{(x - 5)^5} + C$

where $C$ is the constant of integration.

Final Answer

$\boxed{-\frac{1}{4(x - 5)^4} - \frac{1}{(x - 5)^5} + C}$

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

Relative Questions:

1. How do you choose a substitution when integrating a rational function?
2. What are some other common techniques for solving integrals?
3. How do you handle integrals with more complicated polynomial denominators?
4. What is the difference between indefinite and definite integrals?
5. How do you approach integrals involving trigonometric functions?

Tip:

When integrating by substitution, always remember to express all terms in terms of the new variable, including the differential.