$b^{m+n}=b^m \cdot b^n$
$(b^m)^n=b^{m \cdot n}$
$(b \cdot c)^n = b^n \cdot c^n$
+${a^m \div a^n} = {a^{m-n}}$
### $e$ 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
| $f(x)$ | $\int f(x) \cdot dx$ |
| ------------------------------- | ---------------------------- |
-| $k$ (constant) | $kc + c$ |
-| $x^n$ | ${1 \over {n+1}}x^{n+1} + c$ |
-| $1 \over x$ | $\log_e x + c$ |
-| $e^kx$ | ${1 \over k} e^{kx} + c$ |
+| $k$ (constant) | $kx + c$ |
+| $x^n$ | ${1 \over {n+1}}x^{n+1} + c$ |
+| $a x^{-n}$ | $a \cdot \log_e x + c$ |
+| $e^{kx}$ | ${1 \over k} e^{kx} + c$ |
+| $e^k$ | $e^kx + c$ |
| $\sin kx$ | $-{1 \over k} \cos (kx) + c$ |
| $\cos kx$ | ${1 \over k} \sin (kx) + c$ |
| ${f^\prime (x)} \over {f(x)}$ | $\log_e f(x) + c$ |
| $g^\prime(x)\cdot f^\prime(g(x)$ | $f(g(x))$ (chain rule)|
| $f(x) \cdot g(x)$ | $\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx$ |
+| ${1 \over {ax+b}}$ | ${1 \over a} \log_e (ax+b) + c$ |
+| $(ax+b)^n$ | ${1 \over {a(n+1)}}(ax+b)^{n-1} + c$ |
+## Applications of antidifferentiation
+
+- $x$-intercepts of $y=f(x)$ identify $x$-coordinates of stationary points on $y=F(x)$
+- the nature of any stationary point of $y=F(x)$ is determined by the way the sign of the graph of $y=f(x)$ changes about its $x$-intercepts
+- if $f(x)$ is a polynomial of degree $n$, then $F(x)$ has degree $n+1$
+
+To find stationary points of a function, substitute $x$ value of given point into derivative. Solve for ${dy \over dx}=0$. Integrate to find original function.