Mathematics

Evaluate:$$\displaystyle\int{\dfrac{x\,dx}{{x}^{2}+2x+1}}$$


SOLUTION
$$\displaystyle\int{\dfrac{x\,dx}{{x}^{2}+2x+1}}$$
$$=\displaystyle\int{\dfrac{x\,dx}{{\left(x+1\right)}^{2}}}$$
$$=\displaystyle\int{\dfrac{\left(x+1-1\right)\,dx}{{\left(x+1\right)}^{2}}}$$
$$=\displaystyle\int{\dfrac{\left(x+1\right)\,dx}{{\left(x+1\right)}^{2}}}-\displaystyle\int{\dfrac{dx}{{\left(x+1\right)}^{2}}}$$
$$=\displaystyle\int{\dfrac{dx}{x+1}}-\displaystyle\int{{\left(x+1\right)}^{-2}dx}$$
$$=\ln{\left|x+1\right|}-\dfrac{{\left(x+1\right)}^{-2+1}}{-2+1}+c$$
$$=\ln{\left|x+1\right|}+\dfrac{1}{x+1}+c$$
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Subjective Medium Published on 17th 09, 2020
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