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  • Explain solution RD Sharma class 12 chapter 19 Definite Integrals exercise Fill in the blanks question 24 maths

Explain solution RD Sharma class 12 chapter 19 Definite Integrals exercise Fill in the blanks question 24 maths

Answers (1)

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Answer: 5

Hint: Using \int_{0}^{a} f(x) d x=\int_{0}^{a} f(a-x) d x

 

Given: \int_{0}^{10} \frac{x^{10}}{(10-x)^{10}+x^{10}} d x

 

Solution:  

\begin{aligned} Let\; I=&\int_{0}^{10} \frac{x^{10}}{(10-x)^{10}+x^{10}} d x\\ \end{aligned}                    .................(1)

        I=\int_{0}^{10} \frac{(10-x)^{10}}{(10-(10-x))^{10}+(10-x)^{10}} d x

        \mathrm{I}=\int_{0}^{10} \frac{(10-x)^{10}}{x^{10}+(10-x)^{10}} d x        ....................(2)

Adding (1) and (2),

        2I=\int_{0}^{10} \frac{x^{10}}{(10-x)^{10}+x^{10}} d x +\int_{0}^{10} \frac{(10-x)^{10}}{x^{10}+(10-x)^{10}} d x

        2 \mathrm{I}=\int_{0}^{10} \frac{x^{10}+(10-x)^{10}}{x^{10}+(10-x)^{10}} d x

        2 \mathrm{I}=\int_{0}^{10} d x=[x]_{0}^{10}

        \begin{aligned} 2 \mathrm{I}=10-0 \\\\ \Rightarrow \mathrm{I}=5 \end{aligned}

 

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