Calculus Integration

Question

Suppose that  is an even function. If  and , what is ?

Answer 1

Since g(x) is an even function, then int(-a,0) g(x) =int(0,a) g(x) for any a>0. In particular, int(-3,0) g(x)=int(0,3) g(x) =-1. Hence, we have int(-5,0) g(x)=int(-5,-3) g(x) + int(-3,0) g(x) = 4 + (-1)=3. As a result, we have

int(0, 5)g(x) = int(-5, 0)g(x) =3, and so int(-5, 5) g(x) = int(-5,0) g(x) + int(0,5) g(x) =3+3=6.