\(\int_{0}^{1}\displaystyle xe^x\,dx\) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 1
By integration by parts, with x as the first function:
\( \displaystyle \int x e^x\,dx = x e^x - \int 1 \cdot e^x\,dx \)
\( = x e^x - e^x = e^x(x - 1) \)
Applying the limits:
\( \displaystyle \int_{0}^{1} x e^x\,dx = \Big[e^x(x - 1)\Big]_{0}^{1} \)
\( = e(1 - 1) - e^0(0 - 1) \)
\( = 0 + 1 = 1 \)