Question

Prove that `f(x)=x(x/(e^(x)-1)+x/2)` is odd function.

Answer 1

Let `g(x)=(x/(e^(x)-1)+x/2)` then `g(-x)=((-x)/(e^(-x)-1)+(-x)/2)=(x/(e^(x)-1)+x/2)`
`implies g(x)` is even
hence `f(x)=x.g(X)=x(x/(e^(x)-1)+x/2)` is odd function.