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  • Provide solution for RD Sharma maths class 12 chapter 19 Definite Integrals exercise Multiple choice question 27

Provide solution for RD Sharma maths class 12 chapter 19 Definite Integrals exercise Multiple choice question 27

Answers (1)

Answer:

10\left(\frac{\pi}{2}\right)^{9}

Hint:

To solve this equation we convert sin into cos.

Given:

I_{10}=\int_{0}^{\frac{\pi}{2}} x^{10} \sin x \; d x

Solution:
\begin{aligned} &I_{10}=\int_{0}^{\frac{\pi}{2}} x^{10} \sin x\; d x \\\\ &=-x^{10} \cdot \cos x-\int_{0}^{\frac{\pi}{2}} 10 x^{9} \cdot \cos x \; d x \end{aligned}

\begin{aligned} &=0-10 \int_{0}^{\frac{\pi}{2}} x^{9} \cdot \cos x \; d x \\\\ &=10\left[x^{9} \sin x-9 \int_{0}^{\frac{\pi}{2}} x^{8} \sin x \; d x\right]_{0}^{\frac{\pi}{2}} \end{aligned}

\begin{aligned} &=10\left(\frac{\pi}{2}\right)^{9}-90 \int_{0}^{\frac{\pi}{2}} x^{8} \sin x \; d x \\\\ &=10\left(\frac{\pi}{2}\right)^{9}-90 I_{8} \\\\ &=10\left(\frac{\pi}{2}\right)^{9} \end{aligned}

 

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