Discussion
Integrating the logarithm between one and \( e \) gives
Answer: B
Take \( u = \ln x \) and \( dv = dx \), since the logarithm has no elementary antiderivative of its own:
\[ \int_{1}^{e}\ln x\,dx \;=\; \left[x\ln x\right]_{1}^{e} - \int_{1}^{e} dx \;=\; e - (e - 1) \;=\; 1. \]
The general antiderivative is \( x\ln x - x \), which is worth remembering because the logarithm arises so often as one factor of a product
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