6.2.1 Definition & Notation 🔧
$$\int_a^b f(x)\thinspace dx=\lim\limits_{\Delta\to 0}\sum_{k=1}^n f(\bar{x}_k)\cdot \Delta x_k$$
6.2.2 Arithmetic Properties 🔧
Constant Rule
$$\int c\thinspace dx=x+C$$
Constant Multiple Rule
$$\int c\cdot f(x)\thinspace dx=c\cdot\int f(x)\thinspace dx$$
Sum & Difference Rules
$$\int\big(f(x)\pm g(x)\big)\thinspace dx=\int f(x)\thinspace dx\pm \int g(x)\thinspace dx$$
6.2.3 Integration by Parts
$$\int_a^b u\cdot dv=u\cdot v\bigg|_a^b-\int_a^b v\cdot du$$
Proof
Given the
derivative product rule
$$\frac{d}{dx}(u\cdot v)=v\cdot\frac{du}{dx}+u\cdot\frac{dv}{dx}$$
Integrate
$$u\cdot v=\int v\cdot du+\int u\cdot dv$$
Subtract $\int v\cdot du$
6.2.4 Improper Integrals 🔧
