Mathematics

Solve :
$$\displaystyle\int { \cfrac { { x }^{ 2 }+3x-1 }{ { \left( x+1 \right)  }^{ 2 } } dx } $$


SOLUTION
$$\displaystyle\int{\dfrac{\left({x}^{2}+3x-1\right)dx}{{\left(x+1\right)}^{2}}}$$

$$=\displaystyle\int{\dfrac{\left({x}^{2}+x+2x+1-1-1\right)dx}{{\left(x+1\right)}^{2}}}$$

$$=\displaystyle\int{\dfrac{\left({x}^{2}+2x+1+x-2\right)dx}{{\left(x+1\right)}^{2}}}$$

$$=\displaystyle\int{\dfrac{\left({x}^{2}+2x+1\right)dx}{{\left(x+1\right)}^{2}}}+\displaystyle\int{\dfrac{\left(x-2\right)dx}{{\left(x+1\right)}^{2}}}$$

$$=\displaystyle\int{\dfrac{{\left(x+1\right)}^{2}dx}{{\left(x+1\right)}^{2}}}+\displaystyle\int{\dfrac{\left(x+1-3\right)dx}{{\left(x+1\right)}^{2}}}$$

$$=\displaystyle\int{dx}+\displaystyle\int{\dfrac{\left(x+1\right)dx}{{\left(x+1\right)}^{2}}}-3\displaystyle\int{\dfrac{dx}{{\left(x+1\right)}^{2}}}$$

$$=\displaystyle\int{dx}+\displaystyle\int{\dfrac{dx}{\left(x+1\right)}}-3\displaystyle\int{\dfrac{dx}{{\left(x+1\right)}^{2}}}$$

$$=x+\log{\left|x+1\right|}+\dfrac{3}{x+1}+c$$
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Subjective Medium Published on 17th 09, 2020
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