<p>Evaluate \(\displaystyle\int_0^1 x\,e^x\,dx\)</p>
Step-by-Step Solution
Key Concept: IBP: \intx \cdot eˣdx = xeˣ - eˣ + C. Evaluate at limits 0 to 1.
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<p>$\displaystyle\int_0^1 xe^x\,dx = [xe^x]_0^1 - \int_0^1 e^x\,dx$ (IBP with $u=x,\,dv=e^x dx$)</p>
<p>$$= (1\cdot e^1 - 0) - [e^x]_0^1 = e - (e-1) = e - e + 1 = \boxed{1}$$</p>
Correct Answer: A