Questions: ∫ (ln^(2)(3x))/(5x) dx

To solve the integral \(\int \frac{\ln ^{2}(3 x)}{5 x} \, dx\), we can use the substitution method. Let \(u = \ln(3x)\), then \(du = \frac{1}{x} dx\). This will simplify the integral into a more manageable form.

Para resolver la integral \(\int \frac{\ln ^{2}(3 x)}{5 x} \, dx\), utilizamos la sustitución \(u = \ln(3x)\). Entonces, \(du = \frac{1}{x} dx\).

Reescribimos la integral en términos de \(u\): \[ \int \frac{u^2}{5} \, du \]

Integramos la función simplificada: \[ \int \frac{u^2}{5} \, du = \frac{1}{5} \int u^2 \, du = \frac{1}{5} \cdot \frac{u^3}{3} = \frac{u^3}{15} \]

Reemplazamos \(u\) por \(\ln(3x)\): \[ \frac{(\ln(3x))^3}{15} \]

\[ \boxed{\frac{(\ln(3x))^3}{15}} \]