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

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

Answers (1)

Answer:-  Proved

Hints:-  You must know the rules of integration.

Given:-  If  f is an integral function, prove \int_{-a}^{a} f\left(x^{2}\right) d x=2 \int_{0}^{a} f\left(x^{2}\right) d x

Solution:-  f is an integrable function

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

Hence it is an even function

Using integration property, if function is an even function

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

Therefore,

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

Hence proved

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