Mathematics

Evaluate $$\displaystyle \int { \cfrac { 1 }{ \sqrt { x+3 } -\sqrt { x }  }  } dx$$


ANSWER

$$\cfrac{2}{9}{(x+3)}^{3/2}+\cfrac{2}{9}x^{3/2}$$


SOLUTION
Given,

$$\displaystyle \int \dfrac{1}{\sqrt{x+3}-\sqrt{x}}dx$$

$$\displaystyle =\int \dfrac{\sqrt{x+3}+\sqrt{x}}{\left(\sqrt{x+3}\right)^2-\left(\sqrt{x}\right)^2}dx$$   .............  Rationalisation

$$\displaystyle \int \dfrac{1}{3}\left(\sqrt{x}+\sqrt{x+3}\right)dx$$

$$\displaystyle=\dfrac{1}{3}\left(\int \sqrt{x}dx+\int \sqrt{x+3}dx\right)$$

$$\displaystyle=\dfrac{1}{3}\left(\dfrac{2}{3}x^{\tfrac{3}{2}}+\dfrac{2}{3}\left(x+3\right)^{\tfrac{3}{2}}\right)$$

$$\displaystyle=\dfrac{2}{9}\left(x+3\right)^{\tfrac{3}{2}}+\dfrac{2}{9}x^{\tfrac{3}{2}}$$
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Single Correct Medium Published on 17th 09, 2020
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