refinements to calculus notes

diff --git a/spec/calculus.md b/spec/calculus.md
index f98aee74b7997a51b29729e723b9f89cd1f2240a..08cfe8f7a206ba4b2ff1a1ff4d818314215c8e39 100644 (file)
--- a/spec/calculus.md
+++ b/spec/calculus.md
@@ -144,6 +144,7 @@ $\log_b y^{x^n} = x^n \log_b y$
 $b^{m+n}=b^m \cdot b^n$  
 $(b^m)^n=b^{m \cdot n}$  
 $(b \cdot c)^n = b^n \cdot c^n$  
 $b^{m+n}=b^m \cdot b^n$  
 $(b^m)^n=b^{m \cdot n}$  
 $(b \cdot c)^n = b^n \cdot c^n$  

+${a^m \div a^n} = {a^{m-n}}$

 
 ### $e$ as a logarithm
 
 
 ### $e$ as a logarithm
 


@@ -182,7 +183,7 @@ $$\int f(x) dx = F(x) + c$$
 - area enclosed by curves
 - $+c$ should be shown on each step without $\int$
 
 - 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
 
 
 ### Integral laws
 


@@ -191,10 +192,11 @@ $\int k f(x) dx = k \int f(x) dx$
 
 | $f(x)$                          | $\int f(x) \cdot dx$         |
 | ------------------------------- | ---------------------------- |
 
 | $f(x)$                          | $\int f(x) \cdot dx$         |
 | ------------------------------- | ---------------------------- |

-| $k$ (constant)                  | $kc + c$                     |
+| $k$ (constant)                  | $kx + c$                     |

 | $x^n$         | ${1 \over {n+1}}x^{n+1} + c$ |
 | $x^n$         | ${1 \over {n+1}}x^{n+1} + c$ |

-| $1 \over x$ | $\log_e x + c$ |
-| $e^kx$ | ${1 \over k} e^{kx} + c$ |
+| $a \cdot {1 \over x}$ | $a \cdot \log_e x + c$ |
+| $e^{kx}$ | ${1 \over k} e^{kx} + c$ |
+| $e^k$ | $e^kx + c$ |

 | $\sin kx$ | $-{1 \over k} \cos (kx) + c$ |
 | $\cos kx$ | ${1 \over k} \sin (kx) + c$ |
 | ${f^\prime (x)} \over {f(x)}$ | $\log_e f(x) + c$ |
 | $\sin kx$ | $-{1 \over k} \cos (kx) + c$ |
 | $\cos kx$ | ${1 \over k} \sin (kx) + c$ |
 | ${f^\prime (x)} \over {f(x)}$ | $\log_e f(x) + c$ |