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

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

Answers (1)

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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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