Question

If $f (a + b - x) = f (x)$, then $\int\limits^{b}_{{a}}x\, f\, (x) \,dx$ is equal to

• A. $\frac{a+b}{2}\int\limits^{b}_{{a}}f(b-x) \,dx$
• B. $\frac{a+b}{2}\int\limits^{b}_{{a}}f(x) \,dx$
• C. $\frac{b-a}{2}\int\limits^{b}_{{a}}f(x)dx$
• D. $\frac{a+b}{2}\int\limits^{b}_{{a}}f(a+b-x)dx$

Answer 1

Answer: (B)

$\frac{a+b}{2}\int\limits^{b}_{{a}}f(x) \,dx$

Answer 2

$\int\limits^{b}_{{a}}x\, f\, (x) \,dx$ $=\int\limits^{b}_{{a}}(a+b-x) f (a + b - x) dx$.