Q.Evaluate: $\int e^{5x}\cdot\frac{5x\log x+1}{x}\,dx$

Because $\frac{d}{dx}[e^xf(x)]=e^xf(x)+e^xf'(x)=e^x[f(x)+f'(x)]$, any integrand that can be written as $e^x$ multiplied by "some function $f(x)$ plus its own derivative $f'(x)$" integrates immediately to $e^xf(x)+c$, without any explicit by-parts computation. The entire skill lies in recognising — or algebraically manufacturing, by splitting a fraction, factoring a numerator, or substituting an inner variable — the exact pair $f(x)$ and $f'(x)$ hiding inside a more complicated-looking expression multiplying $e^x$ (or, after a subst …