From c852a21dec0421aece1c2b4b9845e9d56d3573bb Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Mon, 6 May 2019 22:03:33 +1000 Subject: [PATCH] [spec] properties of definite integrals --- spec/calculus.md | 14 +++++++++++++- 1 file changed, 13 insertions(+), 1 deletion(-) diff --git a/spec/calculus.md b/spec/calculus.md index b454072..b89eae6 100644 --- 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 $$\int f(u) {du \over dx} \cdot dx = \int f(u) \cdot du$$ @@ -292,7 +304,7 @@ Use identities: - $\cos^2x={1 \over 2}(1+\cos 2x)$ - $\sin 2x = 2 \sin x \cos x$ -### Partial fractions +## Partial fractions On CAS: Action $\rightarrow$ Transformation $\rightarrow$ `expand/combine` -- 2.47.1