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  • Provide solution for RD Sharma maths Class 12 Chapter 19 Definite Integrals Exercise 19.4 (b) Question 33 textbook solution.

Provide solution for  RD Sharma maths Class 12 Chapter 19 Definite Integrals Exercise 19.4 (b) Question 33 textbook solution.

Answers (1)

Answer:-  \frac{16}{15}\sqrt{2}

Hints:-  You must know the rules of integration.

Given:-  \int_{0}^{2} x \sqrt{2-x} \cdot d x

Solution : I=\int_{0}^{2} x \sqrt{2-x} \cdot d x

   I=\int_{0}^{2}(2-x) \sqrt{x} \cdot d x \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \quad\left[\int_{a}^{b} f(x) d x=\int_{a}^{b} f(a+b-x) d x\right]

\begin{aligned} &I=\int_{0}^{2} 2 \sqrt{x} d x-\int_{0}^{2} x^{3 / 2} x \cdot d x \\ &\left.\left.=2 \times \frac{2}{3} x^{3 / 2}\right]_{0}^{2}-\frac{2}{5} x^{5 / 2}\right]_{0}^{2} \\ &I=2 \sqrt{2}\left(\frac{4}{3}-\frac{4}{5}\right) \\ &I=\frac{16}{15} \sqrt{2} \end{aligned}

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