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Provide solution for RD Sharma maths class 12 chapter 19 Definite Integrals exercise Multiple choice question 22

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Answer:

0

Given:

\int_{0}^{1} f(x) d x=1 ; \int_{0}^{1} x f(x) d x=a ; \int_{0}^{1} 2 x^{2} f(x) d x=a^{2}

Hint:

To solve this equation, we have to integrate differentially.
 

Explanation:  

\begin{aligned} &\int_{0}^{1} f(x) d x=1 \ldots(i) \\\\ &\int_{0}^{1} x f(x) d x=a \ldots(i i) \\\\ &\int_{0}^{1} 2 x^{2} f(x) d x=a^{2} \ldots(i i i) \end{aligned}

\begin{aligned} &I=\int_{0}^{1}(a-x)^{2} f(x) d x \\\\ &I=\int_{0}^{1}\left(a^{2}+x^{2}-2 a x\right) f(x) d x \end{aligned}

\begin{aligned} &I=\int_{0}^{1} a^{2} f(x) d x+\int_{0}^{1} x^{2} f(x) d x-\int_{0}^{1} 2 a x f(x) d x \\\\ &I=a^{2} \int_{0}^{1} f(x) d x+\int_{0}^{1} x^{2} f(x) d x-2 a \int_{0}^{1} x f(x) d x \end{aligned}

\begin{aligned} &I=a^{2}+a^{2}-2(a)(a) \\\\ &I=2 a^{2}-2 a^{2} \\\\ &I=0 \end{aligned}

 

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