formulas matematicas

1. FORMULAS DE INTEGRACIÓN
∫0𝑑𝑥 = 𝐶
∫ 𝑘𝑑𝑥 = 𝑘𝑥 + 𝑐
∫ 𝑘𝑓(𝑥)𝑑𝑥 = 𝑘 ∫ 𝑓 (𝑥)𝑑𝑥
∫[𝑓(𝑥) ± 𝑔(𝑥)]𝑑𝑥 = ∫ 𝑓( 𝑥) 𝑑𝑥 ± ∫ 𝑔(𝑥) 𝑑𝑥 + 𝐶
∫ 𝑥 𝑛 𝑑𝑥 =
𝑥 𝑛+1
𝑛+1
+ 𝐶, 𝑛 ≠ −1
∫cos 𝑥 𝑑𝑥 = 𝑠𝑒𝑛 𝑥 + 𝐶
∫ 𝑠𝑒𝑛 𝑥𝑑𝑥 = −cos 𝑥 + 𝐶
∫ 𝑠𝑒𝑐2 𝑥𝑑𝑥 = 𝑡𝑎𝑛𝑥 + 𝐶
∫sec 𝑥 tan 𝑥 𝑑𝑥 = sec 𝑥 + 𝐶
∫ 𝑐𝑠𝑐2 𝑥𝑑𝑥 = − cot 𝑥 + 𝐶
∫csc 𝑥cot 𝑥 𝑑𝑥 = −csc 𝑥 + 𝐶
∫tan 𝑢 𝑑𝑢 = ln|sec 𝑢| + 𝐶
∫cot 𝑢 𝑑𝑢 = ln| 𝑠𝑒𝑛 𝑢| + 𝐶
∫sec 𝑢 𝑑𝑢 = ln| 𝑠𝑒𝑐 𝑢 + tan 𝑢| + 𝐶
∫csc 𝑢 𝑑𝑢 = ln|csc𝑢 − 𝑐𝑜𝑡 𝑢| + 𝐶
∫ 𝑢𝑑𝑢 = 𝑢𝑣 − ∫ 𝑣𝑑𝑢
∫ 𝑢 𝑛 𝑑𝑢 =
1
𝑛+1
𝑢 𝑛+1 + 𝐶, 𝑛 ≠ −1
∫
𝑑𝑢
𝑢
= ln| 𝑢| + 𝐶
∫ 𝑒 𝑢 = 𝑒 𝑢 + 𝐶
∫ 𝑎 𝑢 𝑑𝑢 =
1
ln 𝑎
𝑎 𝑢 + 𝐶
∫
1
𝑥
𝑑𝑥 = ln| 𝑥| + 𝐶
∫
𝑑𝑢
√𝑎2−𝑢2
= 𝑠𝑒𝑛−1 𝑢
𝑎
+ 𝐶
∫
𝑑𝑢
𝑎2+ 𝑢2
=
1
𝑎
𝑡𝑎𝑛−1 𝑢
𝑎
+ 𝐶
∫
𝑑𝑢
𝑢√𝑢2−𝑎2
=
1
𝑎
𝑠𝑒𝑐−1 𝑢
𝑎
+ 𝐶
∫
𝑑𝑢
𝑎2−𝑢2
=
1
2𝑎
ln|
𝑢+𝑎
𝑢−𝑎
| + 𝐶
∫
𝑑𝑢
𝑢2−𝑎2 =
1
2𝑎
ln |
𝑢−𝑎
𝑢+𝑎
|+ 𝐶
Integrales Trascendentales
∫
1
𝑥
𝑑𝑥 =
𝑑𝑥
𝑥
= ln 𝑥 + 𝐶
∫
𝑑𝑥
𝑥 ln 𝑎
= log 𝑎𝑥 + 𝐶
d ∫ 𝑎 𝑥
𝑑𝑥 =
𝑎 𝑥
ln 𝑎
+ 𝐶
∫ 𝑒 𝑥
𝑑𝑥 = 𝑒 𝑥
+ 𝐶
Integrales de Funciones
Trigonométricas Inversas
∫ 𝑠𝑒𝑛−1 𝑢𝑑𝑢 = 𝑢 𝑠𝑒𝑛−1 𝑢 + √1 − 𝑢2 + 𝐶
∫ 𝑐𝑜𝑠−1 𝑢𝑑𝑢 = 𝑢 𝑐𝑜𝑠−1 𝑢 − √1 − 𝑢2 + 𝐶
∫ 𝑡𝑎𝑛−1 𝑢𝑑𝑢 = 𝑢 𝑡𝑎𝑛−1 𝑢 −
1
2
ln(1 + 𝑢2) + 𝐶
∫ 𝑠𝑒𝑐−1 𝑢𝑑𝑢 = 𝑢 𝑠𝑒𝑐−1 𝑢 − ln| 𝑢| + √𝑢2 + 1 + 𝐶
∫ 𝑐𝑠𝑐−1 𝑢𝑑𝑢 = 𝑢 𝑐𝑠𝑐−1 𝑢 − ln| 𝑢 + √𝑢2 − 1| + 𝐶
∫ 𝑐𝑜𝑡−1 𝑢𝑑𝑢 = 𝑢 𝑐𝑜𝑡−1 𝑢 +
1
2
ln(1 + 𝑢2) + 𝐶
Integrales de Funciones Hiperbólicas
Inversas
∫
𝑑𝑢
√ 𝑢2 ±𝑎2
= ln(𝑢 + √𝑢2 ± 𝑎2) + 𝐶
∫
𝑑𝑢
𝑢√ 𝑎2±𝑢2
= −
1
𝑎
ln 𝑎+√ 𝑎2
±𝑢2
)
(𝑢)
+ 𝐶
∫
𝑑𝑢
𝑎2−𝑢2 =
1
2𝑎
ln |
𝑎+𝑢
𝑎−𝑢
| + 𝐶
Directas
∫ 𝑠𝑒𝑛ℎ 𝑥𝑑𝑥 = cosh 𝑥 + 𝐶
∫ cosh 𝑥𝑑𝑥 = 𝑠𝑒𝑛ℎ 𝑥 + 𝐶
∫ tanh 𝑥𝑑𝑥 = 𝑙𝑛 |cosh𝑥| + 𝐶
∫ coth 𝑥𝑑𝑥 = 𝑙𝑛 |senh 𝑥| + 𝐶
∫ sech 𝑥𝑑𝑥 = 𝑡𝑎𝑛−1| 𝑠𝑒𝑛ℎ 𝑥| + 𝐶
∫ csch 𝑥𝑑𝑥 = ln | tanh
𝑥
2
| + 𝐶
∫ sech x tanh 𝑥𝑑𝑥 = 𝑠𝑒𝑐ℎ 𝑥 + 𝐶
∫ 𝑠𝑒𝑛ℎ2
𝑥𝑑𝑥 =
1
4
𝑠𝑒𝑛ℎ 2𝑥 −
𝑥
2
+ 𝐶
∫ 𝑠𝑒𝑐ℎ2
𝑥𝑑𝑥 = 𝑡𝑎𝑛ℎ 𝑥 + 𝐶
∫ 𝑐𝑠𝑐ℎ2
𝑥𝑑𝑥 = −𝑐𝑜𝑡ℎ 𝑥 + 𝐶
∫ csch x coth 𝑥𝑑𝑥 = 𝑐𝑠𝑐ℎ 𝑥 + 𝐶
Integración por Sustitución
Trigonométrica
Si la ∫ 𝑐𝑜𝑛𝑡𝑖𝑒𝑛𝑒 Sustituye Utiliza la identidad
√ 𝑎2 − 𝑢2 u= a sen 𝜃 1−𝑠𝑒𝑛2
𝜃 = 𝑐𝑜𝑠2
𝜃
√ 𝑎2 + 𝑢2 u= a tan 𝜃 1+𝑡𝑎𝑛2
𝜃 = 𝑠𝑒𝑐2
𝜃
√ 𝑢2 − 𝑎2 u= a sec 𝜃 𝑠𝑒𝑐2
𝜃 − 1 = 𝑡𝑎𝑛2
𝜃
Derivada de una Función Logarítmica
Natural
ln( 𝑎𝑏𝑐) = 𝑙𝑛𝑎 + 𝑙𝑛𝑏 + 𝑙𝑛𝑐
ln (
𝑎
𝑏
) = 𝑙𝑛𝑎 − 𝑙𝑛𝑏
ln 𝑎 𝑛
= 𝑛 𝑙𝑛𝑎
Diferenciación Logarítmica
ln |𝑢| = ln|
1
2
𝑑𝑥 𝑈|
Integrales que conducen a Funciones
Logarítmicas
∫ 𝑏 𝑢
𝑑𝑢 =
𝑏 𝑢
ln 𝑏
+ 𝑐, 𝑏 ≠ 1
∫ log 𝑏 𝑢 𝑑𝑢 = ∫
ln 𝑢
ln 𝑏
𝑑𝑢. 𝑏 ≠ 1
Fórmulas de Derivación

2. 𝑢 = 𝑓( 𝑥), 𝑣 = 𝑔( 𝑥), 𝑐 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑒 D=derivada,
C=0
𝑑
𝑑𝑥
[ 𝑘𝑥] = 𝑘
𝑑
𝑑𝑥
[ 𝑘𝑓(𝑥)] = 𝑘𝑓(𝑥)
𝑑
𝑑𝑥
[ 𝑓(𝑥) ± 𝑔(𝑥)] = 𝑓 ′(𝑥) ± 𝑔′(𝑥)
𝑑
𝑑𝑥
[ 𝑥 𝑛] = 𝑛𝑥 𝑛−1
𝐷,( 𝑢 + 𝑣) = 𝐷𝑢 + 𝐷𝑣
𝐷,( 𝑢𝑣) = 𝑢 𝐷𝑣 + 𝑣 𝐷𝑢
𝐷
𝑢
𝑣
=
𝑣 𝐷𝑢−𝑢 𝐷𝑣
𝑣2
𝐷, 𝑓[ 𝑔( 𝑥)] = 𝑓 ′ [ 𝑔( 𝑥)] 𝑔′(𝑥)
𝐷𝑢 𝑛
= 𝑛𝑢 𝑛−1
𝐷 𝑢
𝐷 𝑒 𝑢
= 𝑒 𝑢
𝐷 𝑢
𝐷𝑎 𝑢
= 𝑎 𝑢
ln 𝑎 𝐷 𝑢
𝐷 ln| 𝑢| =
1
𝑢
𝐷 𝑢
𝐷 log 𝑎 𝑢 =
1
𝑢 ln 𝑎
𝐷 𝑢
𝐷 𝑠𝑒𝑛 𝑢 = cos 𝑢 𝐷 𝑢
𝐷 𝑐𝑜𝑠 𝑢 = −𝑠𝑒𝑛 𝑢 𝐷 𝑢
𝐷 𝑡𝑎𝑛 𝑢 = 𝑠𝑒𝑐2
𝑢 𝐷 𝑢
𝐷 𝑐𝑜𝑡 𝑢 = −𝑐𝑠𝑐2
𝑢 𝐷 𝑢
𝐷 𝑠𝑒𝑐 𝑢 = 𝑠𝑒𝑐𝑢 𝑡𝑎𝑛𝑢 𝐷 𝑢
𝐷 𝑐𝑠𝑐 𝑢 = −𝑐𝑠𝑐𝑢 𝑐𝑜𝑡𝑢 𝐷 𝑢
𝐷𝑠𝑒𝑛−1
𝑢 =
1
√1−𝑢2 𝐷 𝑢
𝐷𝑐𝑜𝑠−1
𝑢 =
−1
√1−𝑢2 𝐷 𝑢
𝐷𝑡𝑎𝑛−1
𝑢 =
1
1+𝑢2 𝐷 𝑢
𝐷𝑠𝑒𝑐−1
𝑢 =
1
𝑢√𝑢2−1
𝐷 𝑢
𝐷𝑐𝑜𝑡−1
𝑢 = −
1
1+𝑢2 𝐷 𝑢
𝐷𝑐𝑠𝑐−1
𝑢 = −
1
𝑢√𝑢2 −1
𝐷 𝑢
Identidades Básicas
sen 𝑥 =
1
csc 𝑥
cos 𝑥 =
1
sec 𝑥
sec 𝑥 =
1
cos 𝑥
tan 𝑥 =
𝑠𝑒𝑛 𝑥
cos𝑥
cot 𝑥 =
cos 𝑥
sen 𝑥
csc 𝑥 =
1
sen 𝑥
𝑠𝑒𝑛2
𝜃 + 𝑐𝑜𝑠2
𝜃 = 1
𝑠𝑒𝑛2
𝜃 = 1 − 𝑐𝑜𝑠2
𝜃
𝑐𝑜𝑠2
𝜃 = 1 − 𝑠𝑒𝑛2
𝜃
𝑠𝑒𝑐2
𝜃 = 1 + 𝑡𝑎𝑛2
𝜃
𝑡𝑎𝑛2
𝜃 = 𝑠𝑒𝑐2
𝜃 − 1
1 = 𝑠𝑒𝑐2
𝜃 − 𝑡𝑎𝑛2
𝜃
𝑐𝑠𝑐2
𝜃 = 1 + 𝑐𝑜𝑡2
𝜃
𝑐𝑜𝑡2
𝜃 = 𝑐𝑠𝑐2
𝜃 − 1
1 = 𝑐𝑠𝑐2
𝜃 − 𝑐𝑜𝑡2
𝜃
𝑠𝑒𝑛2
𝑥 = 2𝑠𝑒𝑛 𝑥 cos 𝑥
Identidades Trigonométricas y Funciones
sen 𝜃 =
𝐿𝑂
𝐻
cos 𝜃 =
𝑙𝑎
ℎ
tan 𝜃 =
𝐿𝑂
𝑙𝑎
cot 𝜃 =
𝑙𝑎
𝐿𝑂
csc 𝜃 =
𝐻
𝐿𝑂
sec 𝜃 =
ℎ
𝑙𝑎
Otras identidades
𝑠𝑒𝑛( 𝐴 ± 𝐵) = 𝑠𝑒𝑛𝐴𝑐𝑜𝑠𝐵 ± 𝑐𝑜𝑠𝐴𝑠𝑒𝑛𝐵
𝑐𝑜𝑠( 𝐴 ± 𝐵) = 𝑐𝑜𝑠𝐴𝑐𝑜𝑠𝐵 ± 𝑠𝑒𝑛𝐴𝑠𝑒𝑛𝐵
𝑡𝑎𝑛( 𝐴 ± 𝐵) =
𝑡𝑎𝑛𝐴±𝑡𝑎𝑛𝐵
1±𝑡𝑎𝑛𝐴𝑡𝑎𝑛𝐵
𝑠𝑒𝑛(2𝐴) = 2 𝑠𝑒𝑛 𝐴 𝑐𝑜𝑠 𝐵
𝑠𝑒𝑛(2𝐴) = 𝑐𝑜𝑠2
𝐴 − 𝑠𝑒𝑛2
𝐴
𝑡𝑎𝑛(2𝐴) =
2𝑡𝑎𝑛𝐴
1−𝑡𝑎𝑛2 𝐴
𝑠𝑒𝑛2
𝐴 =
1−𝑐𝑜𝑠2𝐴
2
𝑐𝑜𝑠2
𝐴 =
1+𝑐𝑜𝑠2𝐴
2
𝑡𝑎𝑛2
𝐴 =
1 − 𝑐𝑜𝑠2
𝐴
1 + 𝑐𝑜𝑠2 𝐴
Valores de las Funciones Trigonométricas
0°/360° 30° 45° 60° 90° 180° 270°
Sen 0
1
= 0
1
2
√2
2
√3
2
1 0 −1
Cos 1
1
= 1
√3
2
√2
2
1
2
0 −1 0
Tan 0
1
= 0
√3
3
1 √3 NE 0 NE
Cot 1
0
= 𝑁𝐸 √3 1
√3
3
O NE 0
Sec 1
1
= 1
2√3
3
√2 2 NE −1 NE
Csc 1
0
= 𝑁𝐸 2 √2
2√3
3
1 NE −1