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

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

Answers (1)

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.

 

Posted by

infoexpert23

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