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Evaluate \int_{2}^{3}\frac{\sqrt{x}dx}{\sqrt{5-x}+\sqrt{x}}

  • Option 1)

    1/4    

  • Option 2)

    1/3

  • Option 3)

    1/2

  • Option 4)

    1

 

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As we learnt

Properties of Definite integration -

\int_{a}^{b}f\left ( x \right )dx= \int_{a}^{b}f\left ( a+b-x \right )dx

When \int_{0}^{b}f\left ( x \right )dx= \int_{0}^{b}f\left ( b-x \right )dx

 

- wherein

Put the \left ( a+b-x \right ) at the place of x in f\left ( x \right )

 

 

            Let  I=\int_{2}^{3}\frac{\sqrt{x}dx}{\sqrt{5-x}+\sqrt{x}}

            Using the property \int_{a}^{b}f\left ( x \right )dx= \int_{a}^{b}f\left ( a+b-x \right )dx

we have

I=\int_{2}^{3}\frac{\sqrt{5-x}dx}{\sqrt{5-(5-x)}+\sqrt{5-x}}=\int_{2}^{3}\frac{\sqrt{5-x}dx}{\sqrt{x}+\sqrt{5-x}}

\therefore 2I=\int_{2}^{3}\frac{\sqrt{x}+\sqrt{5-x}dx}{\sqrt{x}+\sqrt{5-x}}=\int_{2}^{3}dx=x\int_{2}^{3}dx=1

\therefore I =\frac{1}{2}

 

 

 


Option 1)

1/4    

Option 2)

1/3

Option 3)

1/2

Option 4)

1

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gaurav

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