Hallar la integral indefinida. \[ \int(u+4)(2 u+1) d(u) \]
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Para resolver la integral indefinida \(\int(u+4)(2u+1) d(u)\), primero expandimos el integrando: \[ (u+4)(2u+1) = 2u^2 + u + 8u + 4 = 2u^2 + 9u + 4. \] Ahora, integramos término a término: \[ \int(2u^2 + 9u + 4) d(u) = \int 2u^2 d(u) + \int 9u d(u) + \int 4 d(u). \] Calculando cada parte: \[ \int 2u^2 d(u) = \frac{2u^3}{3}, \quad \int 9u d(u) = \frac{9u^2}{2}, \quad \int 4 d(u) = 4u. \] Así que uniendo todo, obtenemos: \[ \int(u+4)(2u+1) d(u) = \frac{2u^3}{3} + \frac{9u^2}{2} + 4u + C, \] donde \(C\) es la constante de integración.