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

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

Answers (1)

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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