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  • Need solution for RD Sharma maths class 12 chapter 19 Definite Integrals exercise Fill in the blanks question 23

Need solution for RD Sharma maths class 12 chapter 19 Definite Integrals exercise Fill in the blanks question 23

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Answer: \frac{a}{2}

Hint: Using \int_{0}^{a} f(x) d x=\int_{0}^{a} f(a-x) d x

 

Given: \text { If } \mathrm{f}(\mathrm{x})=\mathrm{f}(\mathrm{a}-\mathrm{x}) \text { and } \int_{0}^{a} x f(x) d x=K \int_{0}^{a} f(x) d x

 

Solution:  f(x)=f(a-x)                    .............(*)

We have  \int_{0}^{a} x f(x) d x=K \int_{0}^{a} f(x) d x

Taking LHS,

        \begin{aligned} &\text { Let } I=\int_{0}^{a} x \end{aligned}                    .............(1)

        \Rightarrow \mathrm{I}=\int_{0}^{a}(a-x) f(a-x) d x

From (*),

        \mathrm{I}=\int_{0}^{a}(a-x) f(x) d x            ...........(2)

Adding (1) and (2),

        2I=\int_{0}^{a} x\; {\mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}\; +} \int_{0}^{a}(a-x) f(x) d x

 

        2I=\int_{0}^{a}(x+a-x) f(x) d x

        \mathrm{I}=\frac{a}{2} \int_{0}^{a} f(x) d x

From the Given, K\int_{0}^{a} f(x) d x =\frac{a}{2} \int_{0}^{a} f(x) d x

On comparing, we get \mathrm{K}=\frac{a}{2}

 

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