4.3 Integration by Parts (부분적분)

$$\frac{d(f(x) \cdot g(x))}{dx} = f(x)\frac{dg(x)}{dx} + g(x)\frac{df(x)}{dx} = f' \cdot g + f \cdot g' = \frac{d(f \cdot g)}{dx}$$

$$f(x) \frac{dg(x)}{dx} = \frac{d(f(x) \cdot g(x))}{dx} - g(x) \frac{df(x)}{dx}$$

$$\int_{}^{}{f(x) \frac{dg(x)}{dx} dx} = \int_{}^{}{\left( \frac{d(f(x) \cdot g(x))}{dx} - g(x) \frac{df(x)}{dx} \right) dx}$$

$$= f(x) \cdot g(x) - \int_{}^{}{g(x) \frac{df(x)}{dx} dx}$$

$$\int_{}^{}{f(x)g'(x)dx} = f(x) \cdot g(x) - \int_{}^{}{g(x)f'(x)dx}$$ $$u = f(x), v = g(x)$$ $$\int_{}^{}{udv} = uv - \int_{}^{}{vdu}$$ $$\int_{a}^{b}{udv} = \left. uv \right|_{a}^{b} - \int_{a}^{b}{vdu}$$