CÁLCULO DIFERENCIAL
CÁLCULO INTEGRAL
𝑑
𝑑𝑥 𝑐 = 0 𝑑
𝑑𝑥 𝑥 = 1
𝑑
𝑑𝑥 𝑙𝑛 𝑣 = 1
𝑣 ∙ 𝑣′
∫(𝑑𝑢+𝑑𝑣−𝑑𝑤)= ∫𝑑𝑢+∫𝑑𝑣−∫𝑑𝑤
∫𝑡𝑎𝑛2𝑣 𝑑𝑣 =tan𝑣− 𝑣+ 𝑐
𝑑
𝑑𝑥 𝑐𝑥 = 𝑐 𝑑
𝑑𝑥 𝑐𝑣 =𝑐∙𝑣′
𝑑
𝑑𝑥 𝑙𝑜𝑔 𝑣 = log𝑒
𝑣 ∙ 𝑣′
∫ 𝑎 𝑑𝑣= 𝑎∫𝑑𝑣
∫ cot 𝑣 𝑑𝑣=ln|𝑠𝑒𝑛 𝑣| + 𝑐
𝑑(𝑢+𝑣−𝑤)
𝑑𝑥 = 𝑢′+𝑣′−𝑤′
𝑑
𝑑𝑥 𝑠𝑒𝑛 𝑣 = cos 𝑣 ∙ 𝑣′
∫𝑑𝑥= 𝑥 + 𝑐 ∫ 𝑥𝑛𝑑𝑥 = 𝑥𝑛+1
𝑛+1+𝑐
∫ sec 𝑣 𝑑𝑣=ln|𝑠𝑒𝑐 𝑣 + tan 𝑣| + 𝑐
𝑑
𝑑𝑥 𝑥 𝑛=𝑛𝑥𝑛−1
𝑑
𝑑𝑥 𝑐𝑜𝑠 𝑣 = −sen 𝑣 ∙ 𝑣′
∫𝑣𝑛𝑑𝑣 = 𝑣𝑛+1
𝑛+1+𝑐
∫ csc 𝑣 𝑑𝑣 =ln|𝑐𝑠𝑐 𝑣 − cot 𝑣 | + 𝑐
𝑑
𝑑𝑥 𝑣 𝑛=𝑛𝑣𝑛−1 ∙ 𝑣′
𝑑
𝑑𝑥 𝑡𝑎𝑛 𝑣 = 𝑠𝑒𝑐2𝑣∙𝑣′
∫𝑑𝑣
𝑣=ln|𝑣|+𝑐
∫ 𝑑𝑣
𝑣2+ 𝑎2= 1
𝑎𝑎𝑟𝑐tan 𝑣
𝑎+𝑐
𝑑
𝑑𝑥 √𝑣 = 1
2 √𝑣 ∙ 𝑣′
𝑑
𝑑𝑥 𝑐𝑜𝑡 𝑣 = −𝑐𝑠𝑐2𝑣∙𝑣′
∫𝑎𝑣𝑑𝑣 = 𝑎𝑣
ln𝑎+𝑐
∫ 𝑑𝑣
𝑣2− 𝑎2=1
2𝑎 ln|𝑣−𝑎
𝑣+𝑎|+𝑐
𝑑
𝑑𝑥 √𝑣
𝑛= 1
𝑛 √𝑣𝑛−1
𝑛 ∙ 𝑣′
𝑑
𝑑𝑥 𝑠𝑒𝑐 𝑣 = sec 𝑣 tan 𝑣 ∙ 𝑣′
∫𝑒𝑣𝑑𝑣 =𝑒𝑣+𝑐
∫ 𝑑𝑣
𝑎2− 𝑣2=1
2𝑎 ln |𝑎+𝑣
𝑎−𝑣|+𝑐
𝑑
𝑑𝑥 (𝑢∙𝑣)=𝑢 ∙𝑣′+𝑣∙𝑢′
𝑑
𝑑𝑥 𝑐𝑠𝑐 𝑣 = −csc 𝑣 cot 𝑣 ∙ 𝑣′
∫ 𝑠𝑒𝑛 𝑣 𝑑𝑣 =−cos𝑣+𝑐
∫𝑑𝑣
√𝑎2−𝑣2 = 𝑎𝑟𝑐 𝑠𝑒𝑛 𝑣
𝑎+𝑐
𝑑
𝑑𝑥 (𝑢
𝑣)= 𝑣∙𝑢′−𝑢 ∙𝑣′
𝑣2
𝑑
𝑑𝑥 𝑎𝑟𝑐 𝑠𝑒𝑛 𝑣 = 1
√1−𝑣2∙𝑣′
∫𝑐𝑜𝑠 𝑣 𝑑𝑣 =sen𝑣+𝑐
∫𝑑𝑣
√𝑣2±𝑎2=𝑙𝑛 (𝑣 + √𝑣2±𝑎2 ) + 𝑐
𝑑
𝑑𝑥 ( 𝑐
𝑣)= − 𝑐
𝑣2 ∙ 𝑣′
𝑑
𝑑𝑥 𝑎𝑟𝑐 𝑐𝑜𝑠 𝑣 = − 1
√1−𝑣2∙𝑣′
∫sec2𝑣 𝑑𝑣= tan𝑣+𝑐
∫√𝑎2−𝑣2 𝑑𝑣 = 𝑣
2√𝑎2−𝑣2+𝑎2
2𝑎𝑟𝑐 𝑠𝑒𝑛 𝑣
𝑎+𝑐
𝑑
𝑑𝑥 (𝑣
𝑐)= 1
𝑐 ∙ 𝑣′
𝑑
𝑑𝑥 𝑎𝑟𝑐 𝑡𝑎𝑛 𝑣 = 1
1+𝑣2∙𝑣′
∫csc2𝑣 𝑑𝑣= −cot𝑣+ 𝑐
∫𝑑𝑣
𝑣√𝑣2−𝑎2 = 1
𝑎𝑎𝑟𝑐sec 𝑣
𝑎+𝑐
𝑑
𝑑𝑥 𝑎𝑣=𝑎𝑣ln𝑎 ∙ 𝑣′
𝑑
𝑑𝑥 𝑎𝑟𝑐 𝑐𝑜𝑡 𝑣 = − 1
1+𝑣2∙𝑣′
∫ 𝑠𝑒𝑐 𝑣 ∙ 𝑡𝑎𝑛 𝑣 𝑑𝑣 =sec𝑣 +𝑐
∫√𝑣2±𝑎2 𝑑𝑣 =𝑣
2√𝑣2±𝑎2±𝑎2
2𝑙𝑛 (𝑣 + √𝑣2±𝑎2 ) + 𝑐
𝑑
𝑑𝑥 𝑒 𝑣=𝑒𝑣 ∙ 𝑣′
𝑑
𝑑𝑥 𝑎𝑟𝑐 𝑠𝑒𝑐 𝑣 = 1
𝑣 √𝑣2−1∙𝑣′
∫ 𝑐𝑠𝑐 𝑣 ∙ 𝑐𝑜𝑡 𝑣 𝑑𝑣 =−csc𝑣+𝑐
Por partes: ∫𝑢𝑑𝑣 =𝑢∙𝑣 −∫𝑣𝑑𝑢 +𝑐
𝑑
𝑑𝑥 𝑢 𝑣=𝑣𝑢𝑣−1∙u′+ln𝑢 ∙ 𝑢𝑣𝑣′
𝑑
𝑑𝑥 𝑎𝑟𝑐 𝑐𝑠𝑐 𝑣 = − 1
𝑣 √𝑣2−1∙𝑣′
∫ tan𝑣 𝑑𝑣 =−ln|cos𝑣|+𝑐= ln|sec𝑣|+𝑐
APLICACIONES DE CÁLCULO DIFERENCIAL
𝑆𝑇= 𝑦
𝑦′
𝑆𝑁=𝑦∙𝑦′
𝑇= 𝑦
𝑦′ √1 + (𝑦′)2
𝑁 = 𝑦 √1 + (𝑦′)2
𝑚 = 𝑝′sen𝜃+𝑝cos𝜃
𝑝′ cos𝜃 − 𝑝 sen 𝜃
tan 𝜃 = 𝑓′(𝑥1) − 𝑔′(𝑥1)
1+𝑓′(𝑥1)∙𝑔′(𝑥1)
𝑚= 𝑔(𝑡)′
ℎ(𝑡)′ = 𝑦′
𝑥′
𝑦 − 𝑦1= 𝑦′(𝑥 − 𝑥1)
𝑦 − 𝑦1=− 1
𝑦′(𝑥 − 𝑥1)
𝛼=𝑥−(𝑦′[1+(𝑦′)2]
𝑦′′ )
𝛽=𝑦+(1+(𝑦′)2
𝑦′′ )
𝑟= √[1 + (𝑦′)2]3
|𝑦"|
𝑟= [√(𝑥′)2+(𝑦′)2]3
|𝑥′ ∙ 𝑦" − 𝑦′ ∙ 𝑥"|
𝑟= [√𝑝 + (𝑝′)2]3
|𝑝2+2(𝑝′)2 − 𝑝 ∙ 𝑝"|
