Mathematics

The value of $$\int _{  }^{  }{ \cfrac { \log { x }  }{ { \left( x+1 \right)  }^{ 2 } }  } dx$$ is


ANSWER

$$\cfrac { -\log { x } }{ x+1 } +\log { x } -\log { \left( x+1 \right) } +C $$


SOLUTION
$$\int \dfrac{lnx}{(x+1)^{2}}$$
$$lnx\int \dfrac{dx}{(x+1)^{2}}-\int \dfrac{1}{x}\int \dfrac{dx}{(x+1)^{2}}$$
$$lnx\dfrac{(x+1)^{-2+1}}{-2+1}-\int \dfrac{1}{x}\dfrac{(x+1)^{-2+1}}{-2+1}dx$$
$$-\dfrac{lnx}{(x+1)}+\int \dfrac{1}{x}\dfrac{1}{(x+1)}dx$$
$$-\dfrac{lnx}{(x+1)}+\int \dfrac{(x+1)-(x)}{x(x+1)}$$
$$-\dfrac{lnx}{(x+1)}+\int \dfrac{1}{x}-\dfrac{1}{1+x}dx$$
$$-\dfrac{lnx}{(1+x)}+lnx-ln(1+x)+C$$
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Single Correct Medium Published on 17th 09, 2020
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