Bijlage A Formules van de goniometrie - overzicht def tan α = def sec α = sin α cos α cot α = 1 cos α cosec α = def cos α sin α def 1 sin α sin2 α + cos2 α = 1 1 + tan2 α = 1 cos2 α 1 + cot2 α = − sin(α ± β) = sin α cos β ± cos α sin β tan(α ± β) = cos(α ± β) = cos α cos β ∓ sin α sin β tan α ± tan β 1 ∓ tan α tan β cot(α ± β) = − 2 tan α 1 − tan2 α sin α = 2t 1 + t2 cos α = 1 − t2 1 + t2 tan α = 2t 1 − t2 sin2 α = 1 − cos(2α) 2 cot(2α) = − met t = tan sin a + sin b = 2 sin a+b 2 2 a+b 2 1 − cot2 α 2 cot α α cos2 α = cos a + cos b = 2 cos 1 ∓ cot α cot β cot α ± cot β cos(2α) = cos2 α − sin2 α sin(2α) = 2 sin α cos α tan(2α) = 1 sin2 α cos a−b 2 cos 1 + cos(2α) 2 a−b 2 sin a − sin b = 2 sin a−b 2 cos a − cos b = −2 sin a−b 2 cos a+b 2 sin a+b 2 2 sin p sin q = sin(p + q) + sin(p − q) 2 cos p sin q = sin(p + q) − sin(p − q) 2 cos p cos q = cos(p + q) + cos(p − q) −2 sin p sin q = cos(p + q) − cos(p − q) 1 Bijlage B Afgeleiden van basisfuncties - overzicht 0 (f + g)0 (x) = f 0 (x) + g 0 (x) ( + 4) = 0 + 40 (f · g)0 (x) = f 0 (x) · g(x) + f (x) · g 0 (x) ( · 4) = 0 · 4 + · 40 0 (f r ) (x) = rf r−1 (x) · f 0 (x) 0 0 (r ) = rr−1 · 0 (r ∈ R) 0 f f 0 (x) · g(x) − f (x) · g 0 (x) (x) = g g(x)2 f (x) = c (c ∈ R) f 0 (x) = 0 f (x) = x f 0 (x) = 1 f (x) = xr (r ∈ R) f (x) = ex f (x) = ax (a ∈ R+ 0 \ {1}) f 0 (x) = rxr−1 4 0 = 0 · 4 − · 40 42 0 (r ) = r · r−1 · 0 0 = e · 0 0 = a ln a · 0 f 0 (x) = ex e f 0 (x) = ax ln a a f (x) = ln x f 0 (x) = 1 x (ln ) = f (x) = a log x f 0 (x) = 1 x ln a ( a log ) = f (x) = sin x f 0 (x) = cos x (sin ) = cos · 0 f (x) = cos x f 0 (x) = − sin x (cos ) = − sin · 0 f (x) = tan x f 0 (x) = 1 cos2 x (tan ) = f (x) = cot x f 0 (x) = −1 sin2 x (cot ) = f (x) = Arcsin x f 0 (x) = √ 1 1 − x2 0 (Arcsin ) = √ f (x) = Arccos x f 0 (x) = √ −1 1 − x2 0 (Arccos ) = √ f (x) = Arctan x f 0 (x) = 1 1 + x2 (Arctan ) = f (x) = Arccot x f 0 (x) = −1 1 + x2 (Arccot ) = 2 0 1 · 0 0 1 · 0 ln a 0 0 1 · 0 cos2 0 0 −1 · 0 sin2 0 0 1 · 0 1 − 2 −1 · 0 1 − 2 1 · 0 1 + 2 −1 · 0 1 + 2 Bijlage C Integralen van basisfuncties - overzicht Z Z sin x = − cos x + c 0 dx = c Z Z dx = x + c Z Z Z xr dx = cos x = sin x + c 1 xr+1 + c (r ∈ R \ {1}) r+1 ex dx = ex + c ax dx = 1 x a ln a 1 dx = ln |x| + c x Z 1 1 dx = √ Arctan k + x2 k Z √ 1 dx = tan x + c cos2 x Z 1 dx = − cot x + c sin2 x Z (a ∈ R+ 0 \ {1}) Z Z Z x √ k Z + c (k > 0) Z p 1 dx = ln x + k + x2 + c k + x2 √ 1 dx = Arcsin x + c = − Arccos x + c 1 − x2 1 dx = Arctan x + c = − Arccot x + c 1 + x2 √ k + x 1 1 dx = √ ln √ + c (k > 0) k − x2 2 k k − x √ 1 dx = Arcsin k − x2 x √ k +c k>0 Z p p 1 p k k + x2 dx = x k + x2 + ln x + k + x2 + c 2 2 Z p 1 p k x k − x2 dx = x k − x2 + Arcsin √ + c (k > 0) 2 2 k Z x 1 dx = ln tan +c sin x 2 Z 1 x + π/2 dx = ln tan +c cos x 2 Z 1 n−1 sin x dx = − sinn−1 x cos x + n n Z 1 n−1 cos x dx = cosn−1 x sin x + n n Z 1 1 sin x n−2 dx = + cosn x n − 1 cosn−1 x n − 1 Z Z lnn x dx = x lnn x − n lnn−1 x dx Z n Z n−2 sin x dx Z 1 1 cos x n−2 1 dx = − + dx sinn x n − 1 sinn−1 x n − 1 sinn−2 x Z Z 1 n xn eax dx = xn eax − xn−1 eax dx (a 6= 0) a a Z Z 3 n cosn−2 x dx 1 dx cosn−2 x