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

Answers (1)


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



\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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