Mathematics

$$\int\dfrac{1}{\sqrt{x}+\sqrt{x+1}}dx$$ is equal to


ANSWER

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


SOLUTION
$$\begin{array}{l} I=\int { \dfrac { 1 }{ { \sqrt { x } +\sqrt { x+1 }  } } dx }  \\I= \int { \dfrac { 1 }{ { \sqrt { x } +\sqrt { x+1 }  } } \times \dfrac { { \sqrt { x } -\sqrt { x+1 }  } }{ { \sqrt { x } -\sqrt { x+1 }  } } dx }  \\I= \int { \dfrac { { \sqrt { x } -\sqrt { x+1 }  } }{ { x-x-1 } } dx }  \\ I=-1\left[ { \int { \sqrt { x } dx } I=-\int { \sqrt { x+1 } dx }  } \right]  \\ I=-\left[ { \dfrac { { { x^{ 3/2 } } } }{ { \dfrac { 3 }{ 2 }  } } -\dfrac { { { { \left( { x+1 } \right)  }^{ 3/2 } } } }{ { \dfrac { 3 }{ 2 }  } }  } \right] +c \\ I=\dfrac { 2 }{ 3 } \left[ { { { \left( { x+1 } \right)  }^{ 3/2 } }-{ x^{ 3/2 } } } \right] +c \end{array}$$

Hence, this is the answer.
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Single Correct Medium Published on 17th 09, 2020
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