Evaluate the following integrals: ( x+1 x )(x+ )^2dx - Vidyadip

Question

Evaluate the following integrals:
$\int\Big(\frac{\text{x}+1}{\text{x}}\Big)(\text{x}+\log\text{x})^2\text{dx}$

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Answer

$\int\Big(\frac{\text{x}+1}{\text{x}}\Big)(\text{x}+\log\text{x})^2\text{dx}$

Let $\text{x}+\log\text{x}=\text{t}$

$\Rightarrow\Big(1+\frac{1}{\text{x}}\Big)=\frac{\text{dt}}{\text{dx}}$

$\Rightarrow\Big(\frac{\text{x}+1}{\text{x}}\Big)\text{dx}=\text{dt}$

Now, $\int\Big(\frac{\text{x}+1}{\text{x}}\Big)(\text{x}+\log\text{x})^2\text{dx}$

$=\int\text{t}^2\text{dt}$

$=\frac{\text{t}^2}{3}+\text{C}$

$=\frac{(\text{x}+\log\text{x})^3}{3}+\text{C}$

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