$b^{m+n}=b^m \cdot b^n$
$(b^m)^n=b^{m \cdot n}$
$(b \cdot c)^n = b^n \cdot c^n$
+${a^m \div a^n} = {a^{m-n}}$
### $e$ as a logarithm
- area enclosed by curves
- $+c$ should be shown on each step without $\int$
-$$\int xn = {x^{n+1} \over n+1} + c$$
+$$\int x^n = {x^{n+1} \over n+1} + c$$
### Integral laws
$\int f(x) + g(x) dx = \int f(x) dx + \int g(x) dx$
$\int k f(x) dx = k \int f(x) dx$
-| $f(x)$ | $\int f(x) \cdot dx$ |
-| ------ | -------------------- |
-| $k$ (constant) | $kc + c$ |
-| $x^n (n \in J\\\{-1\})$ | ${1 \over {n+1}}x^{n+1} + c$ |
+| $f(x)$ | $\int f(x) \cdot dx$ |
+| ------------------------------- | ---------------------------- |
+| $k$ (constant) | $kx + c$ |
+| $x^n$ | ${1 \over {n+1}}x^{n+1} + c$ |
+| $a \cdot {1 \over x}$ | $a \cdot \log_e x + c$ |
+| $e^{kx}$ | ${1 \over k} e^{kx} + c$ |
+| $e^k$ | $e^kx + c$ |
+| $\sin kx$ | $-{1 \over k} \cos (kx) + c$ |
+| $\cos kx$ | ${1 \over k} \sin (kx) + c$ |
+| ${f^\prime (x)} \over {f(x)}$ | $\log_e f(x) + c$ |
+| $g^\prime(x)\cdot f^\prime(g(x)$ | $f(g(x))$ (chain rule)|
+| $f(x) \cdot g(x)$ | $\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx$ |
+