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,
\( \displaystyle \int x e^{x}\,dx = x e^{x} - \int e^{x}\,dx = (x - 1)e^{x} \)
Applying the limits:
\( \displaystyle \int_{0}^{1}x e^{x}\,dx = \Big[(x - 1)e^{x}\Big]_{0}^{1} \)
\( = (0) - (-1) \)
\( = 1 \)