10-Estudo de integrais

Prévia do material em texto

Processing math: 4%
x I F f I F'(x) = f(x)
𝐹 𝑥 𝑓 𝑥 𝐼 𝐹' 𝑥 = 𝑓 𝑥
𝐹 𝑥 = 𝑓 𝑥 = 𝑥
𝐹' 𝑥 = = 𝑥 = 𝑓 𝑥
𝑇 𝑥 = + 9 𝐻 𝑥 = − 2 𝐺 𝑥 = + 𝐶 𝑓 𝑥 = 𝑥
𝑇' 𝑥 = 𝐻' 𝑥 = 𝐺' 𝑥 = 𝑥
𝐹 𝑥 𝑓 𝑥 𝐹 𝑥 + 𝐶 𝑓 𝑥
∫𝑓 𝑥 𝑑𝑥 = 𝐹 𝑥 + 𝐶
∫𝑓 𝑥 𝑑𝑥
𝑓 𝑥
𝑑𝑥
Processing math: 4%
∫𝑓 𝑥 𝑑𝑥 = 𝐹 𝑥 + 𝐶 ↔ 𝐹' 𝑥 = 𝑓 𝑥
∫𝑓 𝑥 𝑑𝑥
∫𝑓 𝑥 𝑑𝑥 = 𝐹 𝑥 + 𝐶 = 𝐹' 𝑥 = 𝑓 𝑥
∫𝑓 𝑥( )𝑑𝑥 = 𝐹 𝑥( ) + 𝐶 → 𝐹 𝑥( ) = 𝐹' 𝑥( ) = 𝑓 𝑥( )
Propriedades da integral indefinida
∫ 𝑘 𝑓 𝑥( )𝑑𝑥 = 𝑘 ∫ 𝑓 𝑥( )𝑑𝑥
∫ 𝑓 𝑥( ) ± 𝑔 𝑥( )( )𝑑𝑥 = ∫ 𝑓 𝑥( )𝑑𝑥 ± ∫ 𝑔 𝑥( )𝑑𝑥
Regra generalizada para integração de uma função
Se 𝑥 é uma função derivável, então: ∫ 𝑥 𝑑𝑥 =
+
+
+ 𝐶, com 𝑛 + 1 ≠ 0.
Processing math: 4%
∫ 7𝑥 + 𝑥 𝑑𝑥 = ∫7𝑥 𝑑𝑥 + ∫𝑥 𝑑𝑥 = 7
+
+
+ 𝐶 +
+
+
+ 𝐶
𝐶 𝐶 𝐶 + 𝐶 = 𝐶 +𝐶
∫ 7𝑥 + 𝑥 𝑑𝑥 = ∫7𝑥 𝑑𝑥 + ∫𝑥 𝑑𝑥 = 7
+
+
+ 𝐶 +
+
+
+ 𝐶 = + + 𝐶
+
+
=
∫𝑥 𝑑𝑥 = + 𝐶
∫1 · 𝑑𝑥 = 𝑥 + 𝐶
Processing math: 4%
∫ 8𝑥 + 4𝑥 − 6𝑥 + 5 𝑑𝑥
∫ 𝑥 + 1 𝑑𝑥
∫ 𝑥 − 𝑥 𝑑𝑥
∫ 𝑥√ 𝑥 − 1 𝑑𝑥
∫
√
𝑑𝑥
𝑦 = 𝑓 𝑥 𝑎 𝑏 𝑆
𝑓 𝑓 𝑥 > 0 𝑥 = 𝑎 𝑥 = 𝑏 𝑥
Figura 10.1 – área S sob a curva contínua f(x), limitada pelas retas x
= a e x = b. Fonte: G. B. Thomas, 2002.
𝑃 𝑎, 𝑏 𝑎, 𝑏 𝑛
Processing math: 4%
𝑎 = 𝑥 < 𝑥 < 𝑥 < … < 𝑥 + < 𝑥 < … < 𝑥 = 𝑏
𝑖 𝑥 − , 𝑥
∆ 𝑥 = 𝑥 − 𝑥 −
𝑥 − 𝑥 − 𝑓 𝑐 𝑐 𝑥 , 𝑥 −
𝑛 𝑆
𝑆 = 𝑓 𝑐 * ∆ 𝑥 + 𝑓 𝑐 * ∆ 𝑥 + … + 𝑓 𝑐 * ∆ 𝑥 = ∑ = 𝑓 𝑐 * ∆ 𝑥
𝑓 𝑐 * ∆ 𝑥 𝑖 = 1 𝑖 = 𝑛
𝑓 𝑃 𝑛
𝑆 𝑃
𝑃||
|
| = 𝑚𝑎𝑥 ∆ 𝑥 ; 𝑖 = 1, 2, 3, …, 𝑛
𝐴 𝑆 𝑓
𝐴 = lim
→
∑ = 𝑓 𝑐 * ∆ 𝑥
𝑓 𝑥 𝑎, 𝑏 𝑃 𝑎, 𝑏 𝑓
𝑥 𝑎, 𝑏 ∫ 𝑓 𝑥 𝑑𝑥
∫ 𝑓 𝑥 𝑑𝑥 = lim
→
∑ = 𝑓 𝑐 * ∆ 𝑥
𝑓 𝑥 𝑑𝑥 𝑎 𝑏
Processing math: 4%
∫ 𝑘 𝑓 𝑥 𝑑𝑥 = 𝑘 ∫ 𝑓 𝑥 𝑑𝑥
∫ 𝑓 𝑥 + 𝑔 𝑥 𝑑𝑥 = ∫ 𝑓 𝑥 𝑑𝑥 + ∫ 𝑔 𝑥 𝑑𝑥
𝑎 < 𝑐 < 𝑏 ∫ 𝑓 𝑥 𝑑𝑥 = ∫ 𝑓 𝑥 𝑑𝑥 + ∫ 𝑓 𝑥 𝑑𝑥
∫ 𝑓 𝑥 𝑑𝑥 = − ∫ 𝑓 𝑥 𝑑𝑥
𝐹 𝐹' 𝑥 = 𝑓 𝑥 𝑥 𝑎, 𝑏
∫ 𝑓 𝑥( )𝑑𝑥 = 𝐹 𝑏( ) − 𝐹 𝑎( )
+𝐶
Processing math: 4%
𝑓 𝑥 = 𝑥
∫ 𝑓 𝑥 𝑑𝑥 = ∫ 𝑥 𝑑𝑥 =
𝑥 𝑥
∫ 𝑓 𝑥 𝑑𝑥 = ∫ 𝑥 𝑑𝑥 = = − = − =
∫ 𝑥 + 3𝑥 − 1 𝑑𝑥 = ∫ 𝑥 𝑑𝑥 + 3∫ 𝑥𝑑𝑥 − ∫ 𝑑𝑥 = + − 𝑥 = + 3 − 0 − + 3 − 1 = 0 + − 1 =
+ −
= =
−1, 0 −
𝑓 𝑥 = 1, 2
∫ 𝑓 𝑥 𝑑𝑥 = ∫ 𝑑𝑥 = ∫ 𝑥− 𝑑𝑥 =
− +
− +
=
−
−
=
−
−
−
−
−
= − + =
Processing math: 4%
𝑓 𝑥 𝑎, 𝑏
𝑓 𝑥 𝑔 𝑥 𝑎, 𝑏
𝐴 𝑓
𝑥
Á𝑟𝑒𝑎 𝑆 = ∫ 𝑓 𝑥 𝑑𝑥
𝑥 = 0 𝑥 = 1 𝑓 𝑥 = 𝑥
á𝑟𝑒𝑎 = ∫ 𝑓 𝑥 𝑑𝑥 = ∫ 𝑥 𝑑𝑥 = = − = 𝑢 . 𝑎 .
𝑢 . 𝑎 .
𝑓 𝑥 = 𝑥 + 1 𝑥 = 0 𝑥 = 4
á𝑟𝑒𝑎 = ∫ 𝑓 𝑥 𝑑𝑥 = ∫ 𝑥 + 1 𝑑𝑥 = + 𝑥 = + 4 − 0 = 8𝑢 . 𝑎 .
𝑓 𝑥 = 5 𝑥 = 1 𝑥 𝑥 = 1 𝑥 = 3
𝑓 𝑥 = 𝑥 𝑥 𝑓 𝑥 = 𝑥 𝑥 = 0 𝑥 = 7
Processing math: 4%
𝑓 𝑥 𝑔 𝑥
𝑎, 𝑏 𝑓 𝑥 > 𝑔 𝑥 𝑥 𝑎, 𝑏 𝑦 = 𝑓 𝑥 𝑦
= 𝑔 𝑥 𝑥 = 𝑎 𝑥 = 𝑏
𝐴 = ∫ 𝑓 𝑥( ) − 𝑔 𝑥( )[ ]𝑑𝑥
Figura 10.2 – Área da região limitada acima por y = f(x), abaixo por
y = g(x), à esquerda pela reta x = a e à direita pela reta x = b. Fonte: G. B.
Thomas, 2002.
𝑦 = 𝑓 𝑥 = 𝑥 + 6 𝑦 = 𝑔 𝑥 = 𝑥 −2, 3
𝐴 = ∫ 𝑓 𝑥 − 𝑔 𝑥 𝑑𝑥𝐴 = ∫− 𝑥 + 6 − 𝑥 𝑑𝑥 = ∫− 𝑥 + 6 − 𝑥 𝑑𝑥𝐴 = ∫− 𝑥𝑑𝑥 + ∫− 6𝑑𝑥 − ∫− 𝑥 𝑑𝑥𝐴 =
+ 6𝑥 −
−
𝐴 = + 6 · 3 − −
−
+ 6 · −2 −
−
𝐴 = + 18 − − − 12 + 𝐴 = − − 𝐴 =
+ =
+
= 𝑢 . 𝑎 .
𝑦 = 𝑓 𝑥 = 𝑥 + 6 𝑦 = 𝑔 𝑥 = 𝑥 −2, 3 𝑢 . 𝑎 .
Processing math: 4%
𝑦 = 1 − 𝑥 𝑦 = 𝑥 − 1 −1, 2
𝑓 𝑔
𝐹 𝑓
𝐹 𝑔 𝑥 𝑓 𝑔 𝑥 𝑔' 𝑥
𝐹 𝑔 𝑥 ' = 𝐹' 𝑔 𝑥 𝑔' 𝑥 = 𝑓 𝑔 𝑥 𝑔' 𝑥
∫𝑓 𝑔 𝑥 𝑔'𝑑𝑥 = 𝐹 𝑔 𝑥 + 𝐶
𝑘 𝑢 = 𝑔 𝑥
∫𝐹' 𝑔 𝑥 𝑔' 𝑥 𝑑𝑥 = ∫ 𝐹 𝑔 𝑥 '𝑑𝑥 = 𝐹 𝑔 𝑥 + 𝐶 = ∫𝐹' 𝑢 𝑑𝑢
𝐹' = 𝑓
∫𝑓 𝑔 𝑥 𝑔' 𝑥 𝑑𝑥 = ∫𝑓 𝑢 𝑑𝑢
Processing math: 4%
∫2𝑥 1 + 𝑥√ 𝑑𝑥 𝑢
𝑢 = 1 + 𝑥 = 2𝑥
𝑑𝑥 =
∫2𝑥 1 + 𝑥√ 𝑑𝑥 = ∫2𝑥 𝑢√
2𝑥 2𝑥
∫2𝑥 𝑢√ = ∫ 𝑢√ 𝑑𝑢
∫ 𝑢√ 𝑑𝑢 = ∫𝑢 𝑑𝑢 = + 𝐶
𝑢 = 1 + 𝑥
∫2𝑥 1 + 𝑥√ 𝑑𝑥 =
+
+ 𝐶
Processing math: 4%
∫ 𝑥 + 3𝑥 · 5𝑥 + 6𝑥 𝑑𝑥
∫ 2𝑥 − 1√ 𝑑𝑥
𝑔' 𝑎, 𝑏 𝑓 𝑢 = 𝑔 𝑥
∫ 𝑓 𝑔 𝑥 𝑔' 𝑥 𝑑𝑥 = ∫ 𝑓 𝑢 𝑑𝑢
Processing math: 4%
∫ 2𝑥 − 1√ 𝑑𝑥
∫ 2𝑥 − 1√ 𝑑𝑥 = ∫ 𝑢√ = ∫ 𝑢 𝑑𝑢 = · 𝑢√
𝑢
𝑢 𝑥
𝑥
∫ 2𝑥 − 1√ 𝑑𝑥 = ∫ 𝑢√ = ∫ 𝑢 𝑑𝑢 = · 𝑢√ = · 2𝑥 − 1 = · 2 · 1 − 1 − · 2 · − 1 = 1√
− 0√ =
∫ 𝑥 3𝑥 + 2√ 𝑑𝑥
∫ 𝑥 𝑥 + 1√ 𝑑𝑥
𝑓 𝑢 = 𝑒
Processing math: 4%
∫𝑒 𝑑𝑢 = 𝑒 + 𝐶
∫𝑒 𝑑𝑥
𝑢 = 5𝑥 = 5𝑑𝑥 =
∫𝑒 𝑑𝑥 = ∫𝑒 = ∫𝑒 𝑑𝑢 = + 𝐶
𝑢 = 5𝑥
∫𝑒 𝑑𝑥 = + 𝐶
𝑓 𝑢 =
∫ = ln 𝑥||
|
| + 𝐶
∫ ∫
+
Processing math: 4%
𝑢 = 3𝑥 = 3𝑑𝑥 =
∫ = ∫ = ∫ = ln 𝑢||
|
| + 𝐶
𝑢 = 3𝑥
∫ = ln 3𝑥||
|
| + 𝐶
+
𝑢 = 𝑥 + 3 = 1𝑑𝑥 = 𝑑𝑢
∫
+
= ∫ = ln 𝑢||
|
| + 𝐶
𝑢 = 𝑥 + 3
∫
+
= ln 𝑥 + 3||
|
| + 𝐶
Processing math: 4%