I see that you've uploaded an image with multiple integrals. I will provide a brief explanation for each integral without a detailed step-by-step process.

1. Integral 5: $\int \frac{x - 1}{x^2 - 4x + 5} \, dx$ Completing the square on the denominator, $x^2 - 4x + 5 = (x-2)^2 + 1$. This transforms it into a standard arctangent integral form: $\int \frac{x - 2}{(x-2)^2 + 1} \, dx$ Resulting in $\ln|(x - 2) + \sqrt{(x-2)^2 + 1}| + C$.

2. Integral 6: $\int \frac{x}{16 + x^4} \, dx$ Use substitution to simplify, $u = x^2$, and then solve the integral in terms of $u$, resulting in an arctangent function.

3. Integral 7: $\int \frac{x}{x^4 + x^2 + 1} \, dx$ Use substitution $u = x^2$, simplifying the expression and solving it as a standard form.

4. Integral 8: $\int \frac{x}{x^2 + 9} \, dx$ This is a direct standard integral: $\frac{1}{2} \ln(x^2 + 9) + C$

5. Integral 9: $\int \frac{x + 3}{x^2 + 9} \, dx$ Split the integral into two parts: $\int \frac{x}{x^2 + 9} \, dx$ and $\int \frac{3}{x^2 + 9} \, dx$. The first part is a standard form, and the second is an arctangent integral.

6. Integral 10: $\int \frac{1}{\sqrt{x^6 - 4}} \, dx$ This integral can be solved through substitution and simplifying the terms, but it involves a more complex technique like trigonometric substitution.

Let me know if you would like further details or step-by-step explanations for any of these integrals.

Further Exploration:

1. How do substitution methods simplify integration?
2. What are the common forms of integrals involving rational functions?
3. How does completing the square help in solving integrals?
4. What are trigonometric substitutions and when are they used?
5. How do logarithmic and arctangent functions relate to integration?

Tip: Remember to check the properties of standard integral forms when solving!