How do you integrate int ln(e^(2x-1))dx?

Jun 26, 2018

${x}^{2} - x + c$

Explanation:

Use knowledge of logarithms

${\log}_{a} {a}^{f} \left(x\right) \equiv f \left(x\right) {\ln}_{a} a \equiv f \left(x\right)$

Hence $\ln {e}^{f} \left(x\right) \equiv f \left(x\right)$

$\implies \int \ln \left({e}^{2 x - 1}\right) \mathrm{dx} = \int \left(2 x - 1\right) \mathrm{dx}$

Use the reverse power rule:

$= {x}^{2} - x + c$