[spec] properties of definite integrals

diff --git a/spec/calculus.md b/spec/calculus.md
index b4540728a25ac9152219336c09153889850573d9..b89eae68a78eddb15321a703067953204a1ade78 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
 
 - *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
 
 $$\int f(u) {du \over dx} \cdot dx = \int f(u) \cdot du$$
 ### Integration by substitution
 
 $$\int f(u) {du \over dx} \cdot dx = \int f(u) \cdot du$$


@@ -292,7 +304,7 @@ Use identities:
 - $\cos^2x={1 \over 2}(1+\cos 2x)$
 - $\sin 2x = 2 \sin x \cos x$
 
 - $\cos^2x={1 \over 2}(1+\cos 2x)$
 - $\sin 2x = 2 \sin x \cos x$
 

-### Partial fractions
+## Partial fractions

 
 On CAS: Action $\rightarrow$ Transformation $\rightarrow$ `expand/combine`
 
 
 On CAS: Action $\rightarrow$ Transformation $\rightarrow$ `expand/combine`