To solve the integral $\int \frac{1}{\left(x+2\right)^2} \, dx$, we can use a simple substitution method.

Step-by-Step Solution:

1. Substitute: Let $u = x + 2$. Then, $du = dx$.

2. Rewrite the Integral: The integral now becomes: $\int \frac{1}{u^2} \, du$

3. Integrate: The integral of $\frac{1}{u^2}$ with respect to $u$ is: $\int u^{-2} \, du = \frac{-1}{u} + C$ where $C$ is the constant of integration.

4. Substitute Back: Now, replace $u$ with $x + 2$ to get the final answer: $\int \frac{1}{\left(x+2\right)^2} \, dx = \frac{-1}{x+2} + C$

Final Answer:

$\int \frac{1}{\left(x+2\right)^2} \, dx = \frac{-1}{x+2} + C$

Would you like a more detailed explanation or have any questions?

Here are 8 related questions to consider:

1. How do you perform substitution in integrals?
2. What is the integral of $\frac{1}{u^2}$?
3. Why is the constant of integration important?
4. How do you reverse substitution in integration problems?
5. Can the substitution method be used for definite integrals?
6. What are other methods to solve integrals involving rational functions?
7. How would you solve the integral $\int \frac{1}{(x+1)^3} \, dx$?
8. What happens if the integrand is $\frac{1}{(x+2)^n}$ for any $n$?

Tip: When performing substitution, always make sure to adjust the limits of integration if working with definite integrals.