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  • Please solve RD Sharma Class 12 Chapter 19 Definite Integrals Exercise 19.4 (b) Question 49 maths textbook solution.

Please solve RD Sharma Class 12 Chapter 19 Definite Integrals Exercise 19.4 (b) Question 49 maths textbook solution.

Answers (1)

Answer:- Proved

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

Given:- Prove  \int_{0}^{\pi} x f(\sin ) d x=\frac{\pi}{2} \int_{0}^{\pi} f(\sin x) \cdot d x

Solution : Let I=\int_{0}^{\pi} x f(\sin ) d x                                 ....(1)

                 \begin{aligned} I &=\int_{0}^{\pi}(\pi-x) f(\sin (\pi-x)) d x \\ &=\int_{0}^{\pi}(\pi-x) f(\sin x) d x \end{aligned}        ......(2)

Adding (1) and (2)

               2 I=\int_{0}^{\pi}(x+\pi-x) f(\sin x) d x

               \begin{aligned} &2 I=\pi \int_{0}^{\pi} f(\sin x) d x \\ &I=\frac{\pi}{2} \int_{0}^{\pi} f(\sin x) d x \\ &\therefore \int_{0}^{\pi} f(\sin x) d x=\frac{\pi}{2} \int_{0}^{\pi} f(\sin x) \cdot d x \end{aligned}

Hence proved.

Posted by

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