Mathematics

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


ANSWER

$$\dfrac{2}{3}[(x+1)^{3/2} + x^{3/2} + c]$$


SOLUTION
Given,

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

multiplying by conjugate, we get,

$$=\int \dfrac{1}{\frac{1}{\sqrt{x+1}+\sqrt{x}}}dx$$

$$=\int \sqrt{x+1}+\sqrt{x}dx$$

$$=\int \sqrt{x+1}dx+\int \sqrt{x}dx$$

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

$$=\dfrac{2}{3}[\left(x+1\right)^{\frac{3}{2}}+x^{\frac{3}{2}}+c]$$
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Single Correct Medium Published on 17th 09, 2020
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