\( \int_{0}^{1} x e^{x} d x \) Area under the curve " \( y^{" \prime} \) from to band above the \( x \) axis

To solve the integral \( \int_{0}^{1} x e^{x} d x \), you can use integration by parts. Letting \( u = x \) and \( dv = e^{x} dx \), we find \( du = dx \) and \( v = e^{x} \). Applying integration by parts gives us: \[ \int x e^{x} dx = x e^{x} - \int e^{x} dx = x e^{x} - e^{x} + C. \] Evaluating this from 0 to 1, we get: \[ \left[ x e^{x} - e^{x} \right]_{0}^{1} = (1 e^{1} - e^{1}) - (0 - 1) = (e - e) + 1 = 1. \] So, the area under the curve from 0 to 1 above the x-axis is equal to 1. It's fascinating to realize that integration by parts is like a dance between two functions, where one leads and the other follows. In this case, \( x \) leads with its linear charm while \( e^x \) twirls gracefully along! For a fun twist, take a moment to visualize the exponential function as a magical growth curve! As it soars, it combines with the linear function \( x \) to create a beautiful ballet of curves. The area under their dance shows how mathematics can intertwine in unexpected ways, revealing that even simple shapes can carry profound meanings!