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Please solve RD Sharma Class 12 Chapter 19 Definite Integrals Exercise 19.4 (b) Question 45 Subquestion (ii) maths textbook solution.

Answers (1)

Answer:-  Proved

Hints:-  You must know the rules of integration.

Given:-  f is an integrable function

Solution:- We have, f is an integrable function

                I=\int_{-a}^{a} x f\left(x^{2}\right) d x


            \begin{aligned} &f(x)=x f\left(x^{2}\right) \\ &f(-x)=-x f\left(x^{2}\right) \\ &=-f(x) \end{aligned}

Hence it is an odd function

Using integration property,

If function is odd, \int_{-a}^{a} f(x) d x=0

           \therefore \int_{-a}^{a} x \cdot f\left(x^{2}\right) d x=0

Hence proved

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