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  • 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

Answers (1)

best_answer

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

Answer (D) is correct

Posted by

Sayak

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