Evaluate the following integrals: (x+2) (x+2)^2 dx - Vidyadip

Question

Evaluate the following integrals:$\int\frac{\log(\text{x}+2)}{(\text{x}+2)^2}\text{dx}$

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Answer

Let $\text{I}=\int\frac{\log(\text{x}+2)}{(\text{x}+2)^2}\text{dx}$
Let $\frac{1}{\text{x}+2}=\text{t}$
$-\frac{1}{(\text{x}+2)^2}\text{dx = dt}$
$\text{I}=-\int\log\big(\frac{1}{\text{t}}\big)\text{dt}$
$=-\int\log\text{t}^{-1}\text{dt}$
$=-\int1\times\log\text{t dt}$
Using integration by parts,
$\text{I}=\log\text{t}\int\text{dt}-\int\big(\frac{1}{\text{t}}\int\text{dt}\big)\text{dt}$
$=\text{t}\log\text{t}-\int\Big(\frac{1}{\text{t}}\times\text{t}\Big)\text{dt}$
$=\text{t}\log\text{t}-\int\text{dt}$
$=\text{t}\log\text{t}-\text{t+C}$
$=\frac{1}{\text{x}+2}\big(\log(\text{x}+2)^{-1}-1\big)+\text{C}$
$\text{I}=\frac{-1}{\text{x}+2}-\frac{\log(\text{x}+2)}{\text{x}+2}+\text{C}$

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