It seems like you're trying to compute the integral of the expression:

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

To simplify this expression, we can break it into two parts:

$\frac{x^2 + 1}{2x} = \frac{x^2}{2x} + \frac{1}{2x}$

Now simplify each term:

$\frac{x^2}{2x} = \frac{x}{2} \quad \text{and} \quad \frac{1}{2x} = \frac{1}{2x}$

Thus, the integral becomes:

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

Now, integrate each term separately:

$\int \frac{x}{2} \, dx = \frac{1}{2} \cdot \frac{x^2}{2} = \frac{x^2}{4}$

$\int \frac{1}{2x} \, dx = \frac{1}{2} \ln|x|$

Therefore, the integral is:

$\int \frac{x^2 + 1}{2x} \, dx = \frac{x^2}{4} + \frac{1}{2} \ln|x| + C$

where $C$ is the constant of integration.

Let me know if you'd like more details or have further questions!

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Related Questions:

1. How do you integrate rational functions that can be simplified into terms of polynomials or fractions?
2. What is the process of simplifying complex fractions before integration?
3. How do logarithmic and power functions appear in integrals?
4. Can you solve similar integrals where the numerator and denominator are quadratic polynomials?
5. What is the general rule for integrating terms of the form $\frac{1}{x}$?

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Tip: Always try to break down complex rational functions into simpler terms before applying the integration rules. It can make the process much easier!