设函数f(x)=x·sinx证明:f(x+2kπ)-f(x)=2kπ·sinx

题目

设函数f(x)=x·sinx证明:f(x+2kπ)-f(x)=2kπ·sinx
如题,x∈R,k∈Z

答案

f(x+2kπ)-f(x)
=(x+2kπ)sin(x+2kπ)-xsinx
=(x+2kπ)sinx-xsinx
=xsinx+2kπsinx-xsinx
=2kπsinx