The value of \(\int_{0}^{1}\displaystyle x\,e^{x}\,dx\) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(1\)
Using integration by parts with x as the first function and \(e^x\) as the second:
\( \displaystyle \int x\,e^{x}\,dx = x\,e^{x} - \int 1 \cdot e^{x}\,dx \)
\(= x\,e^{x} - e^{x} = e^{x}(x - 1)\)
Putting in the limits:
\( \displaystyle \int_{0}^{1} x\,e^{x}\,dx = \Big[e^{x}(x - 1)\Big]_{0}^{1} \)
\(= e(0) - e^{0}(-1)\)
\(= 0 + 1 = 1\)