The value of of $\int\limits_0^1 xe^x\,dx$ is:

1

The correct answer is Option (3) → 1

Evaluate:

$\displaystyle \int_0^1 x e^x \, dx$

Use integration by parts:

Let:

$u = x \quad \Rightarrow \quad du = dx$

$dv = e^x dx \quad \Rightarrow \quad v = e^x$

Then:

$\int x e^x \, dx = x e^x - \int e^x \, dx = x e^x - e^x + C$

Evaluate definite integral:

$\int_0^1 x e^x \, dx = \left[ x e^x - e^x \right]_0^1 = (1 \cdot e^1 - e^1) - (0 \cdot e^0 - e^0) = (e - e) - (0 - 1) = 0 - (-1) = 1$