1. Datos: Integral con raíz lineal en el denominador.
2. Desarrollo:
Sustitución: $u = 3x + 1 \Rightarrow x = \frac{u-1}{3} \Rightarrow dx = \frac{1}{3}du$.
$$ \int \frac{\frac{u-1}{3}}{\sqrt{u}} \frac{1}{3} du = \frac{1}{9} \int \frac{u-1}{u^{1/2}} du = \frac{1}{9} \int (u^{1/2} - u^{-1/2}) du $$
Integrando:
$$ \frac{1}{9} \left[ \frac{2}{3}u^{3/2} - 2u^{1/2} \right] + C $$
Sustituyendo $u = 3x + 1$:
$$ \boxed{\frac{2(3x+1)^{3/2}}{27} - \frac{2(3x+1)^{1/2}}{9} + C} $$