[spec] add to integral laws

diff --git a/spec/calculus.md b/spec/calculus.md
index a085f3b542432402e4b3ba30ffe3514a5433695d..5be32183af996d828aca7dccdb89b11189cc1b06 100644 (file)
--- a/spec/calculus.md
+++ b/spec/calculus.md
@@ -1,3 +1,12 @@
+---
+geometry: margin=2cm
+columns: 2
+graphics: yes
+tables: yes
+author: Andrew Lorimer
+classoption: twocolumn
+---
+

 # Differential calculus
 
 ## Limits
 # Differential calculus
 
 ## Limits


@@ -228,8 +237,10 @@ $\int k f(x) dx = k \int f(x) dx$
 | $f(x)$                          | $\int f(x) \cdot dx$         |
 | ------------------------------- | ---------------------------- |
 | $k$ (constant) | $kx + c$ |
 | $f(x)$                          | $\int f(x) \cdot dx$         |
 | ------------------------------- | ---------------------------- |
 | $k$ (constant) | $kx + c$ |

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

 | $a x^{-n}$ | $a \cdot \log_e x + c$ |
 | $a x^{-n}$ | $a \cdot \log_e x + c$ |

+| ${1 \over {ax+b}}$ | ${1 \over a} \log_e (ax+b) + c$ |
+| $(ax+b)^n$ | ${1 \over {a(n+1)}}(ax+b)^{n-1} + c$ |

 | $e^{kx}$ | ${1 \over k} e^{kx} + c$ |
 | $e^k$ | $e^kx + c$ |
 | $\sin kx$ | $-{1 \over k} \cos (kx) + c$ |
 | $e^{kx}$ | ${1 \over k} e^{kx} + c$ |
 | $e^k$ | $e^kx + c$ |
 | $\sin kx$ | $-{1 \over k} \cos (kx) + c$ |


@@ -237,8 +248,6 @@ $\int k f(x) dx = k \int f(x) dx$
 | ${f^\prime (x)} \over {f(x)}$ | $\log_e f(x) + c$ |
 | $g^\prime(x)\cdot f^\prime(g(x)$ | $f(g(x))$ (chain rule)|
 | $f(x) \cdot g(x)$ | $\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx$ |
 | ${f^\prime (x)} \over {f(x)}$ | $\log_e f(x) + c$ |
 | $g^\prime(x)\cdot f^\prime(g(x)$ | $f(g(x))$ (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
 
 
 ## Applications of antidifferentiation