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  • Please solve RD Sharma Class 12 Chapter 19 Definite Integrals Exercise 19.4 (b) Question 46 maths textbook solution.

Please solve RD Sharma Class 12 Chapter 19 Definite Integrals Exercise 19.4 (b) Question 46 maths textbook solution.

Answers (1)

Answer:-  Proved

Hints:-  You must know the continuity of integral functions.

Given:- f(x) is a continuous on \left [ 0,2a \right ]

Solution : I=\int_{0}^{2 a} f(x) d x

By additive property

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

Consider the integral \int_{0}^{2 a} f(x) d x

\begin{aligned} &\text { Let } x=2 a-t, \text { then } d x=-d t \\ &\text { When } x=a, t=a \quad x=2 a, t=0 \end{aligned}

Hence

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

Therefore,

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

Hence proved.

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