Question:
Evaluate the following integrals:
$\int\left(\frac{x+1}{x}\right)(x+\log x)^{2} d x$
Solution:
Assume $(x+\log x)=t$
$\mathrm{d}(x+\log x)=\mathrm{dt}$
$\Rightarrow\left(1+\frac{1}{\mathrm{x}}\right) \mathrm{dx}=\mathrm{dt}$
$\Rightarrow \frac{x+1}{x} d x=d t$
Substituting $\mathrm{t}$ and $\mathrm{dt}$
$\Rightarrow \int t^{2} d t$
$\Rightarrow \frac{t^{3}}{3}+c$
But $t=x+\log x$
$\Rightarrow \frac{(x+\log x)^{3}}{3}+c$
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