Resuelva
∫e2xsen(3x)dx
Solución

Integración por partes
u=sen(3x)∧dv=e2xdx  du=3cos(3x)dx∧v=e2x2∫e2xsen(3x)dx=e2xse(3x)2−32∫e2xcos(3x)dx(1)
Resolviendo ∫e2xcos(3x)dx
Integración por partes
u=cos(3x)∧dv=e2xdx  du=−3sen(3x)dx∧v=e2x2∫e2xcos(3x)dx=e2xcos(3x)2+32∫e2xsen(3x)dx(2)
Reemplazando (2) en (1)
∫e2xsen(3x)dx=e2xsen(3x)2−32(e2xcos(3x)2+32∫e2xsen(3x)dx)∫e2xsen(3x)dx=e2xsen(3x)2−3e2xcos(3x)4−94∫e2xsen(3x)dx)
Despejamos ∫e2xsen(3x)dx)
∫e2xsen(3x)dx+94∫e2xsen(3x)dx)=e2xsen(3x)2−3e2xcos(3x)444∫e2xsen(3x)dx+94∫e2xsen(3x)dx)=e2xsen(3x)2−3e2xcos(3x)4134∫e2xsen(3x)dx)=e2xsen(3x)2−3e2xcos(3x)4+C∫e2xsen(3x)dx)=4e2xsen(3x)2⋅13−4⋅3e2xcos(3x)13⋅4+C∫e2xsen(3x)dx)=2e2xsen(3x)13−3e2xcos(3x)13+C∫e2xsen(3x)dx)=e2x13(2sen(3x)−3cos(3x))+C

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