We are tasked with solving the following integral:

$\int \frac{x}{x(x^2 + 1)^2} \, dx$

Let's simplify the expression before solving:

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

Now we need to integrate $\int \frac{1}{(x^2 + 1)^2} \, dx$.

Approach:

We'll use the trigonometric substitution: Let $x = \tan(\theta)$, which gives $dx = \sec^2(\theta) d\theta$ and $x^2 + 1 = \sec^2(\theta)$.

Substitute these into the integral:

$\int \frac{1}{(x^2 + 1)^2} \, dx = \int \frac{1}{\sec^4(\theta)} \sec^2(\theta) \, d\theta = \int \cos^2(\theta) \, d\theta$

Now, apply the half-angle identity:

$\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}$

Thus, the integral becomes:

$\int \frac{1 + \cos(2\theta)}{2} \, d\theta = \frac{\theta}{2} + \frac{\sin(2\theta)}{4} + C$

Finally, substitute back $\theta = \tan^{-1}(x)$, and use the identity $\sin(2\theta) = \frac{2x}{x^2 + 1}$ to express everything in terms of $x$:

$\frac{\theta}{2} + \frac{x}{2(x^2 + 1)} + C$

Thus, the solution to the integral is:

$\frac{\tan^{-1}(x)}{2} + \frac{x}{2(x^2 + 1)} + C$

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Would you like more details on any step of this process?

Here are five related questions you might find interesting:

1. How would you integrate $\int \frac{x^2}{(x^2 + 1)^2} \, dx$?
2. Can you use partial fractions to solve integrals involving rational functions?
3. What are common substitutions used for trigonometric integrals?
4. How do we handle improper integrals involving limits?
5. Can this technique be applied to higher powers in the denominator?

Tip: Trigonometric substitutions are very useful for integrals involving quadratic expressions like $x^2 + 1$.