Mathematics

$$I= \int \frac{x+2}{(x+1)^2}dx;$$ then I is equal to 


ANSWER

$$\log (1+x)-\dfrac{1}{x+1}+c$$


SOLUTION
$$\int { \dfrac { x+2 }{ { \left( x+1 \right)  }^{ 2 } }  } dx$$
$$=\int { \dfrac { x+1+1 }{ { \left( x+1 \right)  }^{ 2 } }  } dx$$
$$=\int { \left( \dfrac { x+1 }{ { \left( x+1 \right)  }^{ 2 } } +\dfrac { 1 }{ { \left( x+1 \right)  }^{ 2 } }  \right)  } dx$$
$$=\int { \dfrac { 1 }{ x+1 }  } dx+\int { \dfrac { 1 }{ { \left( x+1 \right)  }^{ 2 } }  } dx$$            $$\left[ \because \int { { x }^{ n }dx=\dfrac { { x }^{ n+1 } }{ n+1 } +c }  \right] $$

$$=\log \left(x+1\right) +\dfrac { { \left( x+1 \right)  }^{ -2+1 } }{ -2+1 } +c$$       $$\left[ \because \int { \dfrac { 1 }{ x } dx=\log { x+c }  }  \right] $$
$$=\log\left(x+1\right) +\left(-1\right) \dfrac{1}{\left(x+1\right)}+c$$
$$=\log\left(x+1\right)-\dfrac{1}{x+1}+c$$
Hence, the answer is $$\log\left(x+1\right)-\dfrac{1}{x+1}+c.$$



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Single Correct Medium Published on 17th 09, 2020
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