\( \begin{array}{l} 67 \int x e^{x} d x \quad \text { ILATE } \\ =u=x \quad d u=e^{x} \\ d u=d x \quad v=e^{x} \\ \int u d u=u-\int u d u \\ \int x e^{x} d x=x\left(e^{x}\right)-\int e^{x} d x \\ =x e^{x}-e^{x}+c \\ =e^{x}[x-1]+c \end{array} \)

Did you know that the technique used here, known as integration by parts, has its roots in Leibniz's work during the 17th century? Leibniz introduced the formula for integration by parts in the context of developing calculus. It allows us to integrate products of functions by reducing them to simpler forms, and it’s like packaging your favorite snacks together for a more satisfying treat! When it comes to applying this integration technique in real life, think about the world of physics and engineering. Many problems, like determining center of mass or solving differential equations, often involve integrals that require methods like integration by parts. So, next time you're solving for the area under a curve in a physics experiment, remember this handy shortcut to keep your calculations flowing smoothly!