The value of \(\int_{0}^{1}\displaystyle x\,e^x\,dx\) is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 1
By integration by parts, with x as the first function and \(e^x\) as the second:
\( \displaystyle \int x\,e^x\,dx = x\,e^x - \int 1 \cdot e^x\,dx = x\,e^x - e^x \)
\( \displaystyle \int_{0}^{1} x\,e^x\,dx = \left[x\,e^x - e^x\right]_{0}^{1} \)
\(= (e - e) - (0 - 1)\)
\(= 1\)