TRIGONOMETRÍA sin(𝛼 + 𝛽) = sin 𝛼 ∙ cos 𝛽 + cos 𝛼 ∙ sin 𝛽 cos(𝛼 + 𝛽) = cos 𝛼 ∙ cos 𝛽 − sin 𝛼 ∙ sin 𝛽 sin2 𝑥 + cos 2 𝑥 = 1 tan2 𝑥 + 1 = sec 2 𝑥 cosh2 𝑥 − sinh2 𝑥 = 1 sin(2𝑥) = 2 sin 𝑥 ∙ cos 𝑥 sin 𝑥 cos 𝑥 cos 𝑥 cot 𝑥 = sin 𝑥 1 sec 𝑥 = cos 𝑥 1 csc 𝑥 = sin 𝑥 sin2 𝑥 = tan 𝑥 = 1 + cos(2𝑥) 2 𝑥 𝑒 − 𝑒 −𝑥 sinh 𝑥 = 2 cos 2 𝑥 = cosh 𝑥 = INTEGRALES DERIVADAS 𝑓(𝑥) = 𝑘 ⟹ 𝑓 ′ (𝑥) = 0 𝑓(𝑥) = 𝑥 ⟹ 𝑓 ′ (𝑥) = 1 𝑓(𝑥) = 𝑐𝑥 ⟹ 𝑓 ′ (𝑥) = 𝑐 𝑓(𝑥) = 𝑥 𝑛 ⟹ 𝑓 ′ (𝑥) = 𝑛 ∙ 𝑥 𝑛−1 𝑓(𝑥) = ln 𝑥 ⟹ 1 𝑓 ′ (𝑥) = 𝑥 𝑓(𝑥) = sin 𝑥 ⟹ 𝑓 ′ (𝑥) = cos 𝑥 𝑓(𝑥) = cos 𝑥 ⟹ 𝑓 ′ (𝑥) = − sin 𝑥 𝑓(𝑥) = 𝑒 𝑥 ∫ 𝑑𝑥 = ∫ 1 𝑑𝑥 = 𝑥 + 𝐶 (regla del producto) ∫ 1 𝑥 ∙ ln 𝛼 ⟹ 𝑓 ′ (𝑥) = 𝑓(𝑥) = tan 𝑥 ⟹ 𝑓 ′ (𝑥) = sec 2 𝑥 𝑓(𝑥) = cot 𝑥 ⟹ 𝑓 ′ (𝑥) = − csc 2 𝑥 𝑓(𝑥) = sec 𝑥 ⟹ 𝑓 ′ (𝑥) = sec 𝑥 ∙ tan 𝑥 𝑓(𝑥) = csc 𝑥 ⟹ 𝑓 ′ (𝑥) = − csc 𝑥 ∙ cot 𝑥 ⟹ 1 𝑑𝑥 = ln|𝑥 | + 𝐶 𝑥 ∫ 𝑒 𝑥 𝑑𝑥 = 𝑒 𝑥 + 𝐶 ∫ sin 𝑥 𝑑𝑥 = − cos 𝑥 + 𝐶 ∫ cos 𝑥 𝑑𝑥 = sin 𝑥 + 𝐶 ∫ 𝛼 𝑥 𝑑𝑥 = 𝛼𝑥 +𝐶 ln 𝛼 ∫ sinh 𝑥 𝑑𝑥 = cosh 𝑥 + 𝐶 ∫ cosh 𝑥 𝑑𝑥 = sinh 𝑥 + 𝐶 𝑓 ′ (𝑥) = 𝛼 𝑥 ∙ ln 𝛼 𝑓(𝑥) = arctan 𝑥 ⟹ 𝑓 ′ (𝑥) = 𝑓(𝑥) = arcsin 𝑥 ⟹ 𝑓 ′ (𝑥) = 𝑓(𝑥) = arccos 𝑥 𝑥 𝑛+1 +𝐶 𝑛+1 𝑓 ′ (𝑥) = 𝑒 𝑥 ⟹ 𝑓(𝑥) = log 𝛼 𝑥 𝑓(𝑥) = 𝛼 𝑥 ∫ 𝑥 𝑛 𝑑𝑥 = ⟹ 1 1 + 𝑥2 1 √1 − 𝑥 2 1 𝑓 ′ (𝑥) = − √1 − 𝑥 2 𝑓(𝑥) = sinh 𝑥 ⟹ 𝑓 ′ (𝑥) = cosh 𝑥 𝑓(𝑥) = cosh 𝑥 ⟹ 𝑓 ′ (𝑥) = sinh 𝑥 ∫ sec 2 𝑥 𝑑𝑥 = tan 𝑥 + 𝐶 ∫ csc 2 𝑥 𝑑𝑥 = − cot 𝑥 + 𝐶 ∫ sec 𝑥 ∙ tan 𝑥 𝑑𝑥 = sec 𝑥 + 𝐶 ∫ csc 𝑥 ∙ cot 𝑥 𝑑𝑥 = − csc 𝑥 + 𝐶 ∫ ∫ 1 𝑑𝑥 = arctan 𝑥 + 𝐶 1 + 𝑥2 1 𝑑𝑥 = arcsin 𝑥 + 𝐶 √1 − 𝑥 2 1 ∫ 𝑑𝑥 = 𝐴𝑟𝑔𝑡𝑎𝑛ℎ 𝑥 + 𝐶 1 − 𝑥2 ∫ ∫ 1 √1 + 𝑥 2 1 √𝑥 2 − 1 1 − cos(2𝑥) 2 𝑑𝑥 = 𝐴𝑟𝑔𝑠𝑖𝑛ℎ 𝑥 + 𝐶 𝑑𝑥 = 𝐴𝑟𝑔𝑐𝑜𝑠ℎ 𝑥 + 𝐶 𝑒 𝑥 + 𝑒 −𝑥 2 LOGARITMOS 𝛼 log𝛼 𝑥 = 𝑥 log 𝛼 1 = 0 log 𝑏𝑛 (𝑎𝑛 ) = log 𝑏 𝑎 log 𝛼 𝛼 = 1 log 𝑏 (𝑎𝑛 ) = 𝑛 ⋅ log 𝑏 𝑎 log 𝛼 (𝑥 ⁄𝑦) = log 𝛼 𝑥 − log 𝛼 𝑦 log 𝛼 (𝑥 ∙ 𝑦) = log 𝛼 𝑥 + log 𝛼 𝑦 log 𝑏 𝑎 = log 𝑏 𝑛 = 𝑥 ⟺ 𝑏 𝑥 = 𝑛 1 log 𝑎 𝑏 log 𝑏 𝑎 = log 𝑐 𝑎 log 𝑐 𝑏 VALOR ABSOLUTO |𝑎| = |−𝑎| |𝑎| 𝑎 | |= |𝑏| 𝑏 |𝑎𝑏| = |𝑎||𝑏| |𝑥 + 𝑦| ≤ |𝑥 | + |𝑦| LÍMITES sin 𝑥 =1 𝑥→0 𝑥 lim lim 1 𝑥 𝑥 tan 𝑥 =1 𝑥→0 𝑥 =1 lim 𝑘 𝑥 𝑥 lim (1 + ) = 𝑒 𝑘 lim (1 + ) = 𝑒 𝑥→∞ 𝑥 𝑥→0 sin 𝑥 𝑥→∞ lim (1 + 𝑥→∞ 𝑘 𝑥+𝑎 ) = 𝑒𝑘 𝑥+𝑎 REGLAS DE DERIVACIÓN 𝑦 = 𝑓(𝑥) ∙ 𝑔(𝑥) 𝑦 ′ = 𝑓 ′ (𝑥) ∙ 𝑔(𝑥) + 𝑓(𝑥) ∙ 𝑔′ (𝑥) ⟹ 𝑦′ = 𝑓 ′ (𝑥) ∙ 𝑔(𝑥) − 𝑓(𝑥) ∙ 𝑔′ (𝑥) 𝑔2 (𝑥) ℎ(𝑥) = 𝑓(𝑔(𝑥)) ⟹ ℎ′ (𝑥) = 𝑓 ′ (𝑔(𝑥)) 𝑔′(𝑥) 𝑓(𝑥) 𝑦= 𝑔(𝑥) Regla de la cadena: ⟹ PROPIEDAD FUNDAMENTAL DE LA DIVISIÓN 𝐷(𝑥) = 𝑑(𝑥) ∙ 𝑐(𝑥) + 𝑅(𝑥) 𝐷(𝑥) 𝑅(𝑥) = 𝑐(𝑥) + 𝑑(𝑥) 𝑑(𝑥) ÁLGEBRA LINEAL Teorema de Laplace: Donde 𝑀𝑖,𝑗 es el determinante de la submatriz obtenida al remover la 𝑖– é𝑠𝑖𝑚𝑎 fila y la 𝑗– é𝑠𝑖𝑚𝑎 columna de 𝐵 . 𝑛 det(𝐵) = ∑(−1)𝑖+𝑗 ⋅ 𝐵𝑖,𝑗 ⋅ 𝑀𝑖,𝑗 𝑗=1 CÓNICAS Para saber el centro (ℎ, 𝑘) sustituir 𝑥 con (𝑥 − ℎ) e 𝑦 con (𝑦 − 𝑘) CIRCUNFERENCIA ELIPSE 𝑥 2 + 𝑦 2 = 𝑟2 𝑥2 𝑦2 + =1 𝑎2 𝑏 2 HIPÉRBOLA 𝑥2 𝑦2 − =1 𝑎2 𝑏 2 𝑦2 𝑥2 − =1 𝑎2 𝑏 2