Integration by parts

\[\begin{align*} \text{aim: } & \int f^{\prime}(x)g(x)dx = f(x)g(x) - \int f(x)g^{\prime}(x)dx\\ \text{set } s(x) &= f(x)g(x)\\ \frac{d}{dx}s(x) &= \lim_{x\to0}\frac{s(x+h) - s(x)}{h} \text{ 根據定義 }\\ &=\lim_{x\to0}\frac{f(x+h)g(x+h) - f(x)g(x)}{h}\\ &= \lim_{x\to0}\frac{f(x+h)g(x+h) -f(x+h)g(x) +f(x+h)g(x) - f(x)g(x)}{h}\\ & = \lim_{x\to0}\frac{f(x+h)g(x+h) -f(x+h)g(x)}{h} + \lim_{x\to0}\frac{f(x+h)g(x) - f(x)g(x)}{h}\\ &= \lim_{x\to0}f(x+h)\frac{g(x+h) - g(x)}{h} + \lim_{x\to0}g(x)\frac{f(x+h) - f(x)}{h}\\ \frac{d}{dx}f(x)g(x) &= f^{\prime}(x)g(x) + f(x)g^{\prime}(x)\\ \text{假若對左} & \text{右兩邊時積分,則會得到: }\\ f(x)g(x) &= \int f^{\prime}(x)g(x)dx + \int f(x)g^{\prime}(x)dx\\ \text{再稍微} & \text{移項一下}\\ \int f^{\prime}(x)g(x)dx &= f(x)g(x) - \int f(x)g^{\prime}(x)dx\\ \text{就是} & \text{"Integration by parts"} \end{align*}\]