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Explain solution for  RD Sharma Class 12 Chapter 19 Definite Integrals Exercise 19.4 (b) Question 44 maths textbook solution.

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Answer:-  Proved

Hints:-  You must know the integration rules of trignometric functions.

Given:-  f(2a-x)=-f(x)

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

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

Using additive property

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

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

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

          \begin{aligned} \therefore \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}

We have f(2a-x)=-f(x)

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

Hence, \int_{0}^{2 a} f(x) d x=0

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