Mathematics

Evaluate: $$\displaystyle\int \frac { 1 } { ( x + 2 ) \sqrt { x + 1 } } d x $$


ANSWER

$$2 \tan ^ { - 1 } ( \sqrt { x + 1 } ) + c$$


SOLUTION
$$I=\displaystyle\int _{  }^{  }{ \frac { 1 }{ { (x+2)\sqrt { x+1 }  } } dx }  $$

$$ Substitute\, u=\sqrt { x+1 }  $$

$$du=\dfrac 1{2\sqrt{x+1}}dx$$

$$=2\displaystyle\int _{  }^{  }{ \dfrac { 1 }{ { 1+{ u^{ 2 } } } } du }  $$

$$ =2{ \tan ^{ -1 }  }u+C $$

$$ =2{ \tan ^{ -1 }  }\left( { \sqrt { x+1 }  } \right) +C$$

Hence the correct option is $$A,\,2{ \tan ^{ -1 }  }\left( { \sqrt { x+1 }  } \right) +C$$
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Single Correct Medium Published on 17th 09, 2020
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