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  • Provide solution for RD Sharma maths class 12 chapter 19 Definite Integrals exercise Multiple choice question 39

Provide solution for RD Sharma maths class 12 chapter 19 Definite Integrals exercise Multiple choice question 39

Answers (1)

Answer:

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

Hint:

To solve this we will split (a+b-x)

Given:

\int(a+b-x)=f(x)

Solution:

\int(a+b-x)=f(x)

Let

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

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

I=\int_{a}^{b}(a+b) \cdot f(x) d x-\int_{a}^{b} x f(x) d x \ldots(i i)

Adding equation (i) and (ii)

I+I=\int_{a}^{b} x f(x) d x+\int_{a}^{b}(a+b) \cdot f(x) d x-\int_{a}^{b} x f(x) d x

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

 

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