[spec] integrals on CAS

diff --git a/spec/calculus.md b/spec/calculus.md
index 01218996b723a7650e38ea770fb8150b31481855..8b4fcb856c09294cff6b5860d463d92626c9a5ac 100644 (file)
--- a/spec/calculus.md
+++ b/spec/calculus.md
@@ -254,6 +254,18 @@ $$\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)$$
 - *Integrand* is $f$.
 - $F(x)$ may be any integral, i.e. $c$ is inconsequential
 
+#### Properties
+
+$$\int^b_a f(x) \> dx = \int^c_a f(x) \> dx + \int^b_c f(x) \> dx$$
+
+$$\int^a_a f(x) \> dx = 0$$
+
+$$\int^b_a k \cdot f(x) \> dx = k \int^b_a f(x) \> dx$$
+
+$$\int^b_a f(x) \pm g(x) \> dx = \int^b_a f(x) \> dx \pm \int^b_a g(x) \> dx$$
+
+$$\int^b_a f(x) \> dx = - \int^a_b f(x) \> dx$$
+
 ### Integration by substitution
 
 $$\int f(u) {du \over dx} \cdot dx = \int f(u) \cdot du$$
@@ -290,8 +302,17 @@ Use identities:
 
 - $\sin^2x={1 \over 2}(1-\cos 2x)$
 - $\cos^2x={1 \over 2}(1+\cos 2x)$
-- $\sin 2x = 2 \sin x \cos x
+- $\sin 2x = 2 \sin x \cos x$
+
+## Partial fractions
 
+On CAS: Action $\rightarrow$ Transformation $\rightarrow$ `expand/combine`  
+or Interactive $\rightarrow$ Transformation $\rightarrow$ `expand` $\rightarrow$ Partial
+
+## Graphing integrals on CAS
+
+In main: Interactive $\rightarrow$ Calculation $\rightarrow$ $\int$ ($\rightarrow$ Definite)  
+Restrictions: `Define f(x)=...` $\rightarrow$ `f(x)|x>1` (e.g.)
 
 ## Applications of antidifferentiation
 
@@ -322,3 +343,11 @@ $$f(x) = {P(x) \over Q(x)} \quad \text{where } P, Q \text{ are polynomial functi
 - when two graphs have the same ordinate, $y$-coordinate is double the ordinate
 - when two graphs have opposite ordinates, $y$-coordinate is 0 i.e. ($x$-intercept)
 - when one of the ordinates is 0, the resulting ordinate is equal to the other ordinate
+
+## Fundamental theorem of calculus
+
+If $f$ is continuous on $[a, b]$, then
+
+$$\int^b_a f(x) \> dx = F(b) - F(a)$$
+
+where $F$ is any antiderivative of $f$