Q

# Choose the correct answers in Exercises 41 to 44. If f( a plus b minus x) equals f( x), then integral a to b x f( x) dx equals to

Choose the correct answers in Exercises 41 to 44.

Q43.    If $f(a+b-x) = f(x)$, then $\int_a^bxf(x)dx$ is equal to

(A)    $\frac{a+b}{2}\int^b_af(b-x)dx$

(B)    $\frac{a+b}{2}\int^b_af(b+x)dx$

(C)    $\frac{b-a}{2}\int^b_af(x)dx$

(D)    $\frac{a+b}{2}\int^b_af(x)dx$

Views

$Let\ \int_a^bxf(x)dx=I$

As we know $\int_a^bf(x)dx=\int_a^bf(a+b-x)dx$

Using the above property we can write the integral as

$\\I=\int_{a}^{b}(a+b-x)f(a+b-x)dx\\ I=\int_{a}^{b}(a+b-x)f(x)dx\\ I=(a+b)\int_{a}^{b}f(x)dx-\int_{a}^{b}xf(x)dx\\ I=(a+b)\int_{a}^{b}f(x)dx-I\\ 2I=(a+b)\int_{a}^{b}f(x)dx\\ I=\frac{a+b}{2}\int_{a}^{b}f(x)dx$