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  • Explain solution for RD Sharma Class 12 Chapter 19 Definite Integrals Exercise 19.4 (a) Question 16 maths textbook solution.

Explain solution for  RD Sharma Class 12 Chapter 19 Definite Integrals Exercise 19.4 (a) Question 16 maths textbook solution.

Answers (1)

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

Given : f(a+b-x)=f(x) and prove that

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

Hint : You must know the formula of \int_{a}^{b} f(x) d x

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

\begin{aligned} &I=\int_{a}^{b}(a+b-x) f(a+b-x) d x\left(\int_{a}^{b} f(x) d x=\int_{a}^{b} f(a+b-x) d x\right) \\ &I=\int_{a}^{b}(a+b-x) f(x) d x[f(a+b-x)=f(x)] \\ &I=\int_{a}^{b}(a+b) f(x) d x-\int_{a}^{b} x f(x) d x \end{aligned}

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

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

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