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Please solve RD Sharma Class 12 Chapter 19 Definite Integrals Exercise 19.4 (b) Question 47 maths textbook solution.

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Answer:- Proved

Hints:-  You must know the rules of integration with its limits.

Given:-  f(a+b-x)=f(x)

Prove \int_{a}^{b} x f(x) d x=\frac{a+b}{2} \int_{a}^{b} f(x) d x

Solution : I=\int_{a}^{b} x f(x) d x                                                                 ......(1)

Using property of definite integral

I=\int_{a}^{b}(a+b-x) \cdot f(a+b-x) d x

It is given that, f(a+b-x)=f(x)

        \begin{aligned} &I=\int_{a}^{b}(a+b-x) \cdot f(x) d x \\ &I=(a+b) \int_{a}^{b} f(x) \cdot d x-\int_{a}^{b} x f(x) d x \end{aligned}                          .....(2)

Adding (1) and (2)

        \begin{aligned} &2 I=(a+b) \int_{a}^{b} f(x) \cdot d x \\ &I=\frac{(a+b)}{2} \int_{a}^{b} f(x) \cdot d x \end{aligned}

Hence proved.


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