WHAT HAPPENS IF…What is the integral from 0 to 1 of xeˣ?
Interactive exploration
Loading the mathematical model…
Connect the picture to the idea∫0bxexdx=beb−∫0bexdx
Since (xeˣ)′=eˣ+xeˣ, integrating both sides from 0 to b gives beᵇ=∫eˣ+∫xeˣ. Rearranging yields (b−1)eᵇ+1. The two bar heights sum to the red boundary-product level.
A detail that matters
The boundary term is evaluated at both endpoints; an indefinite integral also needs a constant.