Resolver \( \int\left(\frac{x^{2}+5 x+1}{x}\right) d x \)
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To solve the integral \( \int\left(\frac{x^{2}+5 x+1}{x}\right) d x \), we first simplify the integrand: \[ \frac{x^{2}+5x+1}{x} = \frac{x^2}{x} + \frac{5x}{x} + \frac{1}{x} = x + 5 + \frac{1}{x} \] Now we can rewrite the integral: \[ \int\left(\frac{x^{2}+5 x+1}{x}\right) d x = \int \left(x + 5 + \frac{1}{x}\right) d x \] Next, we can integrate each term separately: 1. The integral of \( x \) is \( \frac{x^2}{2} \). 2. The integral of \( 5 \) is \( 5x \). 3. The integral of \( \frac{1}{x} \) is \( \ln|x| \). Putting it all together, we get: \[ \int \left(x + 5 + \frac{1}{x}\right) d x = \frac{x^2}{2} + 5x + \ln|x| + C \] Where \( C \) is the constant of integration. Therefore, the final answer is: \[ \int\left(\frac{x^{2}+5 x+1}{x}\right) d x = \frac{x^2}{2} + 5x + \ln|x| + C \]