if  f(a+b-x)=f(x),then\; \int_{a}^{b}x\, f(x)dx\; is \; equal\; to

  • Option 1)

    \frac{a+b}{2}\int_{a}^{b}f(x)dx\;

  • Option 2)

    \;\; \frac{b-a}{2}\int_{a}^{b}f(x)dx\; \;

  • Option 3)

    \; \frac{a+b}{2}\int_{a}^{b}f(a+b-x)dx\;\;

  • Option 4)

    \; \frac{a+b}{2}\int_{a}^{b}f(b-x)dx\;.

 

Answers (1)
V Vakul

As learnt in concept

Properties of Definite integration -

\int_{a}^{b}f\left ( x \right )dx= \int_{a}^{b}f\left ( a+b-x \right )dx

When \int_{0}^{b}f\left ( x \right )dx= \int_{0}^{b}f\left ( b-x \right )dx

 

- wherein

Put the \left ( a+b-x \right ) at the place of x in f\left ( x \right )

 

 f(a+b-x) = f(x)

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

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

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


Option 1)

\frac{a+b}{2}\int_{a}^{b}f(x)dx\;

This is correct

Option 2)

\;\; \frac{b-a}{2}\int_{a}^{b}f(x)dx\; \;

This is incorrect

Option 3)

\; \frac{a+b}{2}\int_{a}^{b}f(a+b-x)dx\;\;

This is incorrect

Option 4)

\; \frac{a+b}{2}\int_{a}^{b}f(b-x)dx\;.

This is incorrect

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