Integration by parts

Statement

Compute $\displaystyle\int_0^1 x\,e^{2x}\,dx$ using integration by parts.

Solution

Let $u=x$, $dv=e^{2x}\,dx$, so $du=dx$, $v=\tfrac12 e^{2x}$: $\int_0^1 x\,e^{2x}\,dx=\Big[\tfrac{x}{2}e^{2x}\Big]_0^1-\int_0^1 \tfrac12 e^{2x}\,dx=\tfrac12 e^2-\Big[\tfrac14 e^{2x}\Big]_0^1=\tfrac12 e^2-\tfrac14 e^2+\tfrac14=\tfrac14(e^2+1).$ Numerically $\tfrac14(7{.}389+1)\approx 2{.}097$.

$\boxed{\int_0^1 x\,e^{2x}\,dx=\tfrac14(e^2+1)\approx 2{.}097}$