Fórmulas de Integración ∫ 0 𝑑𝑥 = 𝐶 ∫ 𝑘𝑑𝑥 = 𝑘𝑥 𝐶 ∫ 𝑘𝑓 (𝑥) 𝑑𝑥 = 𝑘 ∫ 𝑓 (x)𝑑𝑥 ∫[ 𝑓(𝑥) ± 𝑔 (𝑥) ] 𝑑𝑥 = ∫[ 𝑓(𝑥)𝑑𝑥 ± 𝑓 𝑔 (𝑥) 𝑑𝑥 ∫ 𝑥 𝑛 𝑑𝑥 = 𝑥 𝑛+1 𝑛+1 +C,𝑛± − 1 ∫ cos 𝑥𝑑𝑥 = 𝑠𝑒𝑛𝑥 + 𝐶 ∫ 𝑠𝑒𝑛𝑥𝑑𝑥 = - cos 𝑥 + 𝐶 ∫ 𝑠𝑒𝑐 2 𝑥𝑑𝑥 = tan 𝑥 + 𝐶 ∫ 𝑠𝑒𝑐 𝑥 𝑡𝑎𝑛 𝑥𝑑𝑥 = 𝑠𝑒𝑐 𝑥 + 𝐶 ∫ 𝑐𝑠𝑒 2 𝑥𝑑𝑥 = - cot 𝑥 + 𝐶 ∫ 𝑐𝑠𝑒 𝑥 𝑐𝑜𝑡 𝑥𝑑𝑥 = - 𝑐𝑠𝑐 x + 𝐶 ∫ 𝑡𝑎𝑛 𝑢𝑑𝑢 = 𝐼𝑛|𝑠𝑒𝑐 𝑢| + 𝐶 ∫ cot 𝑢𝑑𝑢 = 𝐼𝑛|𝑠𝑒𝑛𝑢| + 𝐶 ∫ 𝑠𝑒𝑐 𝑢𝑑𝑢 = 𝐼𝑛|𝑠𝑒𝑐 𝑢 + 𝑡𝑎𝑛 𝑢| + 𝐶 ∫ csc 𝑢𝑑𝑢 = 𝐼𝑛|𝑐𝑠𝑐 u + 𝑐𝑜𝑡 u| +𝐶 ∫ 𝑢𝑑𝑢 = 𝑢𝑣 - ∫ 𝑣𝑑𝑢 1 ∫ 𝑢𝑛 𝑑𝑢 = 𝑛+1 𝑢𝑛+1 + C, n ± 1 ∫ 𝑑𝑢 𝑢 == 𝐼𝑛|u|+ C ∫ 𝑒 ′′ = 𝑒 ′′ + 𝐶 ′′ ∫ 𝑎′′ 𝑑𝑢 = 1 𝐼𝑛 𝑎 𝑎′′ + 𝐶 1 ∫ 𝑥 𝑑𝑥 = 𝐼𝑛 + 𝐶 𝑑𝑢 𝑢 ∫ √𝑎2−𝑢2 = 𝑠𝑒𝑛−1 𝑎 + 𝐶 𝑑𝑢 𝑢 ∫ √𝑎2−𝑢2 = 𝑡𝑎𝑛−1 𝑎 + 𝐶 𝑑𝑢 𝑢 ∫ √𝑎2−𝑢2 = 𝑠𝑒𝑐 −1 𝑎 + 𝐶 𝑑𝑢 1 𝑢+𝑎 ∫ √𝑎2−𝑢2 = 𝑎|𝑢−𝑎| + 𝐶 𝑑𝑢 1 𝑢−𝑎 ∫ 𝑢2 −𝑎2 = 2𝑎 𝐼𝑛 |𝑢+𝑎| + 𝐶 Integrales Trascendentes 1 𝑑𝑥 ∫ 𝑥 𝑑𝑥 = ( 𝑥 ) = 𝐼𝑛 |𝑥| + 𝐶 𝑑𝑥 ∫ 𝑥 𝐼𝑛 𝑎 = 𝑙𝑜𝑔 𝑎𝑥 + 𝐶 𝑎𝑥 𝑑 ∫ 𝑎 𝑥 𝑑𝑥 = 𝐼𝑛 𝑎 + 𝐶 ∫ 𝑒 𝑥 𝑑𝑥 = 𝑒 𝑥 + 𝐶 Integrales de Funciones Trigonométricas Inversas ∫ 𝑠𝑒𝑛−1 𝑢𝑑𝑢 = 𝑢𝑠𝑒𝑛 −1 u + √1 − 𝑢2 + 𝐶 ∫ 𝑐𝑜𝑠 −1 𝑢𝑑𝑢 = u 𝑐𝑜𝑠 −1 u - √1 − 𝑢2 + 𝐶 1 ∫ 𝑡𝑎𝑛−1 𝑢𝑑𝑢 = u - 2 𝐼𝑛 ( 1+ 𝑢2 + 𝐶 ∫ 𝑠𝑒𝑐 −1 𝑢𝑑𝑢 = 𝑢 𝑠𝑒𝑐 −1 – 𝐼𝑛 |𝑢|+ √𝑢2 + 1 + 𝐶 ∫ 𝑐𝑠𝑐 −1 𝑢𝑑𝑢 = 𝑢 𝑐𝑠𝑐 −1 u - 𝑖𝑛|𝑢 + √𝑢2 – 1 | + 𝐶 1 ∫ 𝑐𝑜𝑡 −1 𝑢𝑑𝑢 = u 𝑐𝑜𝑡 −1 u + 2 𝐼𝑛( 1 + 𝑢2 ) + 𝐶 Integrales de Funciones Hiperbólicas Inversas: ∫√ 𝑑𝑢 𝑢2 ± 𝑎 2 = In ( u + √𝑢2 ± 𝑎2 ) + 𝐶 1 𝐼𝑛 𝑎 + √𝑎2 ± 𝑢2 𝑑𝑢 ∫ 𝑢 √𝑎2± 𝑢2 = 𝑎 ∫ 𝑑𝑢 𝑎 2 − 𝑢2 (𝑢) 1 + 𝐶 𝑎+𝑢 = 2𝑎 𝐼𝑛 |𝑎−𝑢| + 𝐶 Directas ∫ 𝑠𝑒𝑛 ℎ𝑥𝑑𝑥 = 𝑐𝑜𝑠ℎ 𝑥 + 𝐶 ∫ cosh 𝑥𝑑𝑥 = 𝑠𝑒𝑛ℎ𝑥 + 𝐶 ∫ 𝑡𝑎𝑛ℎ 𝑥𝑑𝑥 = 𝐼𝑛|𝑐𝑜𝑠ℎ x| + 𝐶 ∫ 𝑐𝑜𝑡ℎ 𝑥𝑑𝑥 = 𝐼𝑛 |𝑠𝑒𝑛ℎ𝑥| + 𝐶 ∫ 𝑠𝑒𝑐 ℎ𝑥𝑑𝑥 =𝑡𝑎𝑛−1 |𝑠𝑒𝑛ℎ𝑥| + 𝐶 𝑥 ∫ 𝑐𝑠𝑐 ℎ𝑥𝑑𝑥 = 𝐼𝑛|𝑡𝑎𝑛ℎ 2|+ 𝐶 ∫ 𝑠𝑒𝑐 ℎ𝑥 tanh 𝑥𝑑𝑥 = sec hx + 𝐶 1 𝑥 ∫ 𝑠𝑒𝑛ℎ2 𝑥𝑑𝑥 = 4 senh2x - 2 + 𝐶 ∫ 𝑠𝑒𝑐 ℎ2 𝑥𝑑𝑥 = 𝑡𝑎𝑛ℎ 𝑥 + 𝐶 ∫ 𝑐𝑠𝑐 ℎ2 𝑥𝑑𝑥= - 𝑐𝑜𝑡ℎ + 𝐶 ∫ 𝑐𝑠𝑐 ℎ𝑥 coth 𝑥 = 𝑐𝑠𝑐 ℎ𝑥 + 𝐶 Integración Por Sustitución Trigonométrica Si la Integral Contiene Se Sustituye Utiliza la Identidad √𝑎2 − 𝑢2 u = asen θ 1 - 𝑠𝑒𝑛2 = 𝑐𝑜𝑠 2 θ √𝑎2 − 𝑢2 u = a tan θ 1 + 𝑡𝑎𝑛2 θ 𝑠𝑒𝑐 2 θ √𝑢2 − 𝑎2 u = a 𝑠𝑒𝑐 θ 𝑠𝑒𝑐 2 – 1 = 𝑡𝑎𝑛2 θ Formulas de Derivación u= 𝑓 (𝑥), 𝑣 = 𝑔(𝑥), 𝐶 𝑒𝑠 𝑢𝑛𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑒 𝐷, 𝐶 = 0 𝑑 𝑑𝑥 𝑑 𝑑𝑥 𝑑 𝑑𝑥 𝑑 𝑑𝑥 [𝑘𝑥] = 𝑘 [𝑘𝑓(𝑥)] = 𝑘𝑓 (𝑥) [𝑓(𝑥) ± 𝑔(𝑥)] = 𝑓′(𝑥) ± 𝑔′(𝑥) [𝑥 ′′ ] = 𝑛𝑥 𝐷, (𝑢 + 𝑣) = 𝐷, 𝑢+𝐷, 𝑣 𝐷, (𝑢𝑣) = 𝑢𝐷, 𝑣 + 𝑣𝐷, 𝑢 𝑢 𝐷, 𝑣 = 𝑣𝐷,𝑢−𝑢𝐷,𝑣 𝑣 2 𝐷, 𝑓(𝑔)𝑥)) = 𝑓'(𝑔(𝑥))𝑔'(𝑥) 𝐷, 𝑢𝑛 = 𝑛𝑢𝑛−1 D,u 𝐷,𝑒 𝑢 =𝑒 𝑢 D, u 𝐷, 𝑎𝑢 = 𝑎𝑢 𝐼𝑛𝑎𝐷, 𝑢 1 𝐷, In |u|= 𝑢 𝐷, 𝑢 1 𝐷, 𝑙𝑜𝑔𝑎 |u|= 𝑢 𝐼𝑛 𝑎 𝐷, 𝑢 𝐷, 𝑠𝑒𝑛𝑢 = 𝑐𝑜𝑠𝑢𝐷, 𝑢 𝐷, 𝑐𝑜𝑠𝑢= -𝑠𝑒𝑛𝑢𝐷, 𝑢 𝐷, 𝑡𝑎𝑛𝑢= 𝑠𝑒𝑐 2 𝑢𝐷, 𝑢 𝐷, 𝑐𝑜𝑡𝑢= -𝑐𝑠𝑐 2 𝑢𝐷, 𝑢 𝐷, 𝑠𝑒𝑐𝑢= 𝑠𝑒𝑐𝑢 𝑡𝑎𝑛𝑢𝐷, 𝑢 𝐷, 𝑐𝑠𝑐𝑢= -𝑐𝑠𝑐𝑢 𝑐𝑜𝑡𝑢𝐷, 𝑢 1 1) 𝐷,𝑠𝑒𝑛−1 u= √1−𝑢2 𝐷, 𝑢 −1 2) 𝐷, 𝑐𝑜𝑠 −1 u= 1−𝑢2 𝐷, 𝑢 1 3) 𝐷, 𝑡𝑎𝑛−1 u = 1+𝑢2 𝐷, 𝑢 4) 𝐷, 𝑠𝑒𝑐 −1 u = 1 𝑢√𝑢2 𝐷, 𝑢 1 5) 𝐷𝑥 (𝑐𝑜𝑡)−1 u = - 1+𝑢2 . u' 6) 𝐷𝑥 (𝑐𝑠𝑐)−1 u = - Identidades Básicas 1 𝑆𝑒𝑛𝑥 = 𝑐𝑠𝑐 𝑥 1 𝐶𝑜𝑠 𝑥 = 𝑠𝑒𝑐 𝑥 1 𝑆𝑐𝑐 𝑥 = 𝑐𝑜𝑠 𝑥 𝑠𝑒𝑛𝑥 𝑇𝑎𝑛 𝑥 = 𝑐𝑜𝑠 𝑥 𝑐𝑜𝑠 𝑥 𝐶𝑜𝑡 𝑥 = 𝑠𝑒𝑛𝑥 𝐶𝑠𝑐 𝑥 = 1 𝑠𝑒𝑛𝑥 1 𝑢√𝑢2 −1 . u' 𝑠𝑒𝑛2 θ 𝑐𝑜𝑠 2 = 1 𝑠𝑒𝑛2 θ= 1 - 𝑐𝑜𝑠 2 θ 𝑐𝑜𝑠 2 θ= 1 - 𝑠𝑒𝑛2 θ 1 + 𝑡𝑎𝑛2 θ 𝑠𝑒𝑐 2 θ => 𝑡𝑎𝑛2 θ = 𝑠𝑒𝑐 2 θ – 1 1 + 𝑐𝑜𝑡 2 θ = 𝑐𝑠𝑐 2 θ=> 𝑐𝑜𝑡 2 θ = 𝑐𝑠𝑐 2 θ – 1 𝑆𝑒𝑛2𝑥= 2𝑠𝑒𝑛𝑥𝑐𝑜𝑠 𝑥 Cos2 θ = 2 𝑠𝑒𝑛2 θ – 1 Cos 2 θ = 1 – 2 𝑐𝑜𝑠 2 θ 2𝑡𝑎𝑛θ Tan 2 θ = 1+𝑡𝑎𝑛2 θ 2 𝑐𝑜𝑠 2 θ = 1- cos 2 θ 𝑐𝑜𝑠 2 θ = 1 − 𝑐𝑜𝑠 2 θ 2 Identidades Trigonométricas y Funciones Sen θ = Cos θ = 𝐶.𝑂 𝐻 𝐶.𝐴 𝐻 𝐶.𝑂 Tan θ = 𝐶.𝐴 𝐻 Csc θ = 𝐶.𝑂 𝐻 Sec θ = 𝑆.𝐴 𝐶.𝐴 Cot θ = 𝐶.𝑂 1 𝑠𝑒𝑛θ = csc θ Sen θ = 1 𝑐𝑜𝑠 θ 1 𝑐𝑠𝑐 θ = sec θ 1 Cos θ = 𝑠𝑒𝑐 θ 1 Tan θ = 𝑐𝑜𝑡 θ