∵函数f(x)=ex(sinx-cosx), ∴f′(x)=(ex)′(sinx-cosx)+ex(sinx-cosx)′=2exsinx, ∵x∈(2kπ,2kπ+π)(k∈Z)时,f′(x)>0,x∈(2kπ+π,2kπ+2π)(k∈Z)时,f′(x)<0, ∴x∈(2kπ,2kπ+π)(k∈Z)时f(x)递增,x∈(2kπ+π,2kπ+2π)(k∈Z)时f(x)递减, ∴当x=2kπ+π(k∈Z)时,f(x)取极大值, 其极大值为f(2kπ+π)=e2kπ+π[sin(2kπ+π)-cos(2kπ+π)] =e2kπ+π×(0-(-1)) =e2kπ+π, 又x∈(0,2013π), ∴函数f(x)的各极大值之和S=eπ+e3π+e5π+…+e2011π ═=. 故选:B. |