It is well known that cooperation cannot be an evolutionary stable strategy for a non-iterative game in a well-mixed population. In contrast, structured populations favor cooperation since cooperators can benefit each other by forming local clusters. Especially, previous studies have shown that scale-free networks strongly promote cooperation. However, little is known about the invasion mechanism of cooperation in scale-free networks. Here we conducted evolutionary simulations of the evolution of cooperation in scale-free networks where, starting from all defectors, cooperators can spontaneously emerge by mutation. The purpose of this study is to reveal microscopic and macroscopic behaviors of the cooperators' invasion into the network. Since the evolutionary dynamics are influenced by the definition of fitness, we tested two commonly adopted fitness functions: accumulated payoff and average payoff. The simulation results show that cooperation is strongly enhanced in the case of the accumulated payoff when the temptation for defection is low. However, as the temptation becomes higher, cooperation persists more in the case of the average payoff. Moreover, in the case of the average payoff, lower degree nodes play a more important role in spreading cooperative strategies compared to the case of the accumulated payoff.