$\int_0^1 x e^x dx$ is equal to

Integration by parts is used when integrating the product of functions.

The formula is:

$\int u \, dv = uv - \int v \, du$

Choose:

• $u = x$
• $dv = e^x dx$

Then:

• $du = dx$
• $v = e^x$

---

$\int x e^x dx = x \cdot e^x - \int e^x \cdot dx$

$= x e^x - e^x$

$= e^x(x - 1)$

---

Applying the limits from 0 to 1:

$\int_0^1 x e^x dx = [e^x(x-1)]_0^1$

At $x = 1$:

$e^1(1-1) = 0$

At $x = 0$:

$e^0(0-1) = -1$

---

$\int_0^1 x e^x dx = 0 - (-1)$

$= 1$

Therefore, the answer is $1$.