FormularioCI

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Formulário - Cálculo Integral 
 
� 𝑓(𝑥)𝑏
𝑎
𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎) 𝑠𝑒𝑛𝑑𝑜 𝐹´ = 𝑓 � 𝐹´(𝑥)𝑏
𝑎
 𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎) �𝑢𝑑𝑣 = 𝑢𝑣 − �𝑣 𝑑𝑢 
𝑉 = � 𝜋𝑟2𝑏
𝑎
𝑑𝑥 𝑜𝑢 𝑉 = � 𝜋𝑟2𝑑
𝑐
𝑑𝑦 𝑓𝑚𝑒𝑑 = 1𝑏 − 𝑎� 𝑓(𝑥)𝑏𝑎 𝑑𝑥 �𝑘𝑑𝑥 = 𝑘𝑥 + 𝑐 
�𝑥𝑛𝑑𝑥 = 𝑥𝑛+1
𝑛 + 1 + 𝑐 𝑛 ≠ −1 �𝑒𝑥𝑑𝑥 = 𝑒𝑥 + 𝑐 � 1𝑥 𝑑𝑥 = ln|𝑥| + 𝑐 
�𝑠𝑒𝑛 𝑥 𝑑𝑥 = −𝑐𝑜𝑠𝑥 + 𝑐 �𝑐𝑜𝑠 𝑥 𝑑𝑥 = 𝑠𝑒𝑛𝑥 + 𝑐 � sec2 𝑥 𝑑𝑥 = 𝑡𝑔𝑥 + 𝑐 
�𝑐𝑜𝑠 sec2 𝑥 𝑑𝑥 = −𝑐𝑜𝑡𝑔𝑥 + 𝑐 � sec 𝑥 𝑡𝑔 𝑥 𝑑𝑥 = 𝑠𝑒𝑐𝑥 + 𝑐 �𝑐𝑜𝑠𝑠𝑒𝑐𝒙 𝑐𝑜𝑡𝑔𝒙 𝑑𝑥 = −𝑐𝑜𝑠𝑠𝑒𝑐 𝑥 + 𝑐 
 
Expressão Substituição Identidade 
�𝑎2 − 𝑥2 𝑥 = 𝑎 𝑠𝑒𝑛𝜃 𝑐𝑜𝑠2𝜃 = 1 − 𝑠𝑒𝑛2𝜃 
�𝑎2 + 𝑥2 𝑥 = 𝑎 𝑡𝑔𝜃 𝑠𝑒𝑐2𝜃 = 1 + 𝑡𝑔2𝜃 
�𝑥2 − 𝑎2 𝑥 = 𝑎 𝑠𝑒𝑐𝜃 𝑡𝑔2𝜃 = 𝑠𝑒𝑐2𝜃 − 1 
 
𝑠𝑒𝑛2(𝑥) + cos2(𝑥) = 1 sec2(𝑥) = 1 + 𝑡𝑔2(𝑥) 𝑐𝑜𝑠𝑠𝑒𝑐2(𝑥) = 1 + 𝑐𝑜𝑡𝑔2(𝑥) 
𝑠𝑒𝑛2𝑥 = 12 (1 − cos 2𝑥) 𝑐𝑜𝑠2𝑥 = 12 (1 + cos 2𝑥) 𝑐𝑜𝑠(𝑎 + 𝑏) = cos 𝑎 cos 𝑏 − 𝑠𝑒𝑛 𝑎 𝑠𝑒𝑛 𝑏 𝑐𝑜𝑠(𝑎 − 𝑏) = cos 𝑎 cos 𝑏 + 𝑠𝑒𝑛 𝑎 𝑠𝑒𝑛 𝑏 𝑠𝑒𝑛(𝑎 + 𝑏) = sen 𝑎 cos 𝑏 + 𝑠𝑒𝑛 𝑏 cos 𝑎 𝑠𝑒𝑛(𝑎 − 𝑏) = sen 𝑎 cos 𝑏 − 𝑠𝑒𝑛 𝑏 cos 𝑎 𝑐𝑜𝑠(2𝑎) = cos2 𝑎 − 𝑠𝑒𝑛2 𝑎 𝑠𝑒𝑛(2𝑎) = 2sen𝑎 cos 𝑎 
 
 |𝑃1𝑃2| = �(𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)2 + (𝑧2 − 𝑧1)2 𝑟2 = (𝑥 − 𝑥𝑐)2 + (𝑦 − 𝑦𝑐)2 + (𝑧 − 𝑧𝑐)2 
 
𝑟 = 𝑟0���⃗ + 𝑡�⃗� 𝑥 = 𝑥0 + 𝑎𝑡 𝑦 = 𝑦0 + 𝑏𝑡 𝑧 = 𝑧0 + 𝑐𝑡 𝑥 − 𝑥0
𝑎
= 𝑦 − 𝑦0
𝑏
= 𝑧 − 𝑧0
𝑐
 
 
𝑎(𝑥 − 𝑥0) + 𝑏(𝑦 − 𝑦0) + 𝑐(𝑧 − 𝑧0) = 0 𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 + 𝑑 = 0 𝑜𝑛𝑑𝑒 𝑑 = −(𝑎𝑥0 + 𝑏𝑦0 + 𝑐𝑧0) 
 
 
𝑆𝑒 𝑧 = 𝑓(𝑥,𝑦)𝑒 𝑥 = 𝑔(𝑡) e 𝑦 = ℎ(𝑡): → 𝑑𝑧
𝑑𝑡
= 𝜕𝑧
𝜕𝑥
 𝑑𝑥
𝑑𝑡
 + 𝜕𝑧 𝜕𝑦 𝑑𝑦𝑑𝑡 
𝑆𝑒 𝑧 = 𝑓(𝑥,𝑦)e 𝑥 = 𝑔(𝑠, 𝑡) e 𝑦 = ℎ(𝑠, 𝑡) : → 𝜕𝑧
𝜕𝑠
= 𝜕𝑧
𝜕𝑥
 𝜕𝑥
𝜕𝑠
 + 𝜕𝑧 𝜕𝑦 𝜕𝑦𝜕𝑠 e 𝜕𝑧𝜕𝑡 = 𝜕𝑧𝜕𝑥 𝜕𝑥𝜕𝑡 + 𝜕𝑧 𝜕𝑦 𝜕𝑦𝜕𝑡 
 
𝑑𝑧
𝑑𝑥
= −𝐹𝑥
𝐹𝑧
 𝑑𝑧
𝑑𝑦
= −𝐹𝑦
𝐹𝑧
 
 
𝐷𝑢𝑓(𝑥, 𝑦) = 𝑓𝑥(𝑥,𝑦)𝑎 + 𝑓𝑦(𝑥, 𝑦)𝑏 𝐷𝑢𝑓(𝑥, 𝑦, 𝑧) = 𝑓𝑥(𝑥, 𝑦, 𝑧)𝑎 + 𝑓𝑦(𝑥,𝑦, 𝑧)𝑏 + 𝑓𝑧(𝑥, 𝑦, 𝑧)𝑐 
 
∇𝑓(𝑥,𝑦) = 〈�𝑓𝑥(𝑥,𝑦),𝑓𝑦(𝑥,𝑦)〉 = 𝜕𝑓𝜕𝑥 𝚤� + 𝜕𝑓𝜕𝑦 𝚥 
 
∇𝑓(𝑥,𝑦, 𝑧) = 〈�𝑓𝑥(𝑥, 𝑦, 𝑧), 𝑓𝑦(𝑥, 𝑦, 𝑧),𝑓𝑧(𝑥, 𝑦, 𝑧)〉 = 𝜕𝑓𝜕𝑥 𝚤� + 𝜕𝑓𝜕𝑦 𝚥 + 𝜕𝑓𝜕𝑧 𝑘�⃗ 
 
 
 
𝐷 = 𝐷(𝑎, 𝑏) = 𝑓𝑥𝑥(𝑎, 𝑏)𝑓𝑦𝑦(𝑎, 𝑏) − [𝑓𝑥𝑦(𝑎, 𝑏)]2 
 
 
𝐷 = �𝑓𝑥𝑥 𝑓𝑥𝑦𝑓𝑦𝑥 𝑓𝑦𝑦�