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  • Provide Solution for RD Sharma Class 12 Chapter 19 Definite Integrals Exercise Revision Exercise Question 39

Provide Solution for RD Sharma Class 12 Chapter 19 Definite Integrals Exercise Revision Exercise Question 39

Answers (1)

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Answer:  \frac{a}{2}

Hint: To solve this equation use formula  \int_{0}^{a} f(x) d x

Given:

\int_{0}^{a} f(x) d x

Solution:

\begin{aligned} &\int_{0}^{a} f(x) d x=\int_{0}^{a} f(a-x) d x\\ & \end{aligned}

I=\int_{0}^{a} \frac{\sqrt{a-x}}{\sqrt{a-x}+\sqrt{a-(a-x)}} d x\\                   ........(1)

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

Adding (1) and (2)

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

\begin{aligned} &2 I=\int_{0}^{a} \frac{\sqrt{x}+\sqrt{a-x}}{\sqrt{a+x}+\sqrt{x}} d x \\ & \end{aligned}

2 I=\int_{0}^{a} d x \\

2 I=(x)_{0}^{a}   

\begin{aligned} &\quad \Rightarrow a-0=a \\ & \end{aligned}

2 I=a \\

I=\frac{a}{2}

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