Concept: Definite Integrals: a∫bf(x)dx=a∫bf(a+b−x)dx If f(x)=f(2a−x), then 0∫2af(x)dx=20∫af(x)dx A function f(x) is: - Even, if f(−x)=f(x). And −a∫af(x)dx=20∫af(x)dx. - Odd, if f(−x)=−f(x). And −a∫af(x)dx=0. - Periodic, if f(np±x)=f(x), for some number p and n∈Z. Calculation: We know that for an even function, f(−x)=f(x) and −a∫af(x)dx=20∫af(x)dx. For the given condition 0∫a[f(x)+f(−x)]dx=−a∫ag(x)dx to be true, f(x) and g(x) both must be even functions, i.e. f(x)=f(−x) and g(x)=f(x)+f(−x).