The integral you provided is:

$\int \frac{x+3}{(x^2 + 5)^2} \, dx$

Solution Strategy

This problem can be solved without using partial fraction decomposition, as it is not suited for that method given the presence of the squared denominator. Instead, we can use the method of substitution and consider splitting the integral based on a numerator manipulation strategy.

1. Substitute: Let $u = x^2 + 5$, which gives us $du = 2x \, dx$. This substitution simplifies the integral.

2. Integral Splitting: Since the numerator is $x + 3$, we can split the integral as follows:

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

   For the first integral, using the substitution $u = x^2 + 5$ and for the second integral, applying a standard integral formula or a trigonometric substitution should help evaluate the solution.

Would you like me to continue with a detailed solution using this method or explore a different approach?

Related Questions:

1. How would the approach change if the denominator was not squared?
2. Can we use trigonometric substitutions to simplify $\int \frac{1}{(x^2 + 5)^2} \, dx$?
3. How would we handle this integral if it had $(x^2 - 5)$ in the denominator?
4. What is the importance of substitution in solving integrals like this?
5. How would the integral be evaluated if the numerator was a higher-degree polynomial?

Tip: For integrals involving quadratics, always consider substitution and trigonometric identities to simplify the expressions before deciding on other methods like partial fractions.