Mathematics

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

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

##### 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
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

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