Text Compute ∫01x ex dx\displaystyle\int_0^1 x\,e^{x}\,dx∫01xexdx by integration by parts. Solution Let u=xu=xu=x, dv=exdxdv=e^{x}dxdv=exdx, so du=dxdu=dxdu=dx, v=exv=e^{x}v=ex. ∫01xexdx=[xex]01−∫01exdx=e−(e−1)=1\displaystyle\int_0^1 x e^{x}dx=\big[x e^{x}\big]_0^1-\int_0^1 e^{x}dx=e-(e-1)=1∫01xexdx=[xex]01−∫01exdx=e−(e−1)=1. I=1\boxed{I=1}I=1