DERIVADAS
● 𝑑
𝑑𝑥 (C) = 0
● 𝑑
𝑑𝑥 (X) = 1
● 𝑑
𝑑𝑥 (C*V) = C* 𝑑
𝑑𝑥 (V)
● 𝑑
𝑑𝑥 (U*V) = VU´+UV´
● 𝑑
𝑑𝑥 ( 𝑈
𝑉 ) = 𝑉𝑈´−𝑈𝑉´
𝑉2
● 𝑑
𝑑𝑥 (𝑉𝑛) = n*𝑉𝑛−1 ∗𝑑
𝑑𝑥 (𝑣 )
● 𝑑
𝑑𝑥 (𝑋𝑛) = n*𝑋𝑛−1
● 𝑑
𝑑𝑥 (U+V-W) = U´+V´-W´
● 𝑑
𝑑𝑥 (𝑎𝑣) = 𝑎𝑣∗ln𝑎 ∗ 𝑑
𝑑𝑥 (𝑣)
INTEGRALES ORDINARIAS 1-7
1) ∫𝑑𝑥 = 𝑥 + 𝐶
2) ∫𝐶 ∗ 𝑑𝑥 = 𝐶 ∫𝑑𝑥
3) ∫𝑑𝑢+𝑑𝑣−𝑑𝑤 =∫𝑑𝑢+∫𝑑𝑣−∫𝑑𝑤
4) ∫𝑉𝑛∗𝑑𝑣 =𝑉𝑛+1
𝑛+1 +𝐶
5) ∫𝑒𝑣∗𝑑𝑣 = 𝑒𝑣+ 𝐶
6) ∫𝑎𝑣∗𝑑𝑣 =𝑎𝑣
ln𝑎+𝐶
7) ∫𝑑𝑣
𝑣=ln𝑣 + 𝐶
DERIVADA FUNCIONES TRIGONOMETRICAS
● 𝑑
𝑑𝑥 (sen v) = cos v * 𝑑
𝑑𝑥 (v)
● 𝑑
𝑑𝑥 (cos v) = −𝑠𝑒𝑛 v * 𝑑
𝑑𝑥 (v)
● 𝑑
𝑑𝑥 (tg v) = 𝑠𝑒𝑐2v * 𝑑
𝑑𝑥 (v)
● 𝑑
𝑑𝑥 (ctg v) = −𝑐𝑠𝑐2v * 𝑑
𝑑𝑥 (v)
● 𝑑
𝑑𝑥 (sec v) = sec v * tg v * 𝑑
𝑑𝑥 (v)
● 𝑑
𝑑𝑥 (csc v) = −𝑐𝑠𝑐 v * ctg v * 𝑑
𝑑𝑥 (v)
IDENTIDADES TRIGONOMETRICAS
csc𝐴 = 1
𝑠𝑒𝑛 𝐴 sec 𝐴 = 1
𝑐𝑜𝑠 𝐴
sen𝐴 = 1
𝑐𝑠𝑐 𝐴 cos𝐴 = 1
𝑠𝑒𝑐 𝐴
tg𝐴 = 1
𝑐𝑡𝑔 𝐴 tg𝐴 = 𝑠𝑒𝑛 𝐴
𝑐𝑜𝑠 𝐴
ctg𝐴 = 1
𝑡𝑔 𝐴 ctg𝐴 = cos𝐴
𝑠𝑒𝑛 𝐴
INTEGRALES ORDINARIAS 8-17
8) ∫𝑠𝑒𝑛 𝑣 𝑑𝑣 = − cos𝑣 + 𝐶
9) ∫𝑐𝑜𝑠 𝑣 𝑑𝑣 = 𝑠𝑒𝑛 𝑣 + 𝐶
10) ∫𝑡𝑔 𝑣 𝑑𝑣 = −𝑙𝑛(cos𝑣) + 𝐶 / 𝑙𝑛(sec𝑣) + 𝐶
11) ∫𝑐𝑡𝑔 𝑣 𝑑𝑣 =𝑙𝑛(sen𝑣) + 𝐶
12) ∫𝑠𝑒𝑐 𝑣 𝑑𝑣 =𝑙𝑛 (sec𝑣 + 𝑡𝑔 𝑣) + 𝐶
13) ∫𝑐𝑠𝑐 𝑣 𝑑𝑣 =𝑙𝑛 (csc 𝑣 − 𝑐𝑡𝑔 𝑣) + 𝐶
14) ∫𝑠𝑒𝑐 2𝑣 𝑑𝑣 =𝑡𝑔 𝑣 + 𝐶
15) ∫𝑐𝑠𝑐 2𝑣 𝑑𝑣 = −𝑐𝑡𝑔 𝑣 + 𝐶
16) ∫𝑠𝑒𝑐 𝑣 𝑡𝑔 𝑣 𝑑𝑣 = 𝑠𝑒𝑐 𝑣 + 𝐶
17) ∫𝑐𝑠𝑐 𝑣 𝑐𝑡𝑔 𝑣 𝑑𝑣 = −𝑐𝑠𝑐 𝑣 + 𝐶
FORMULAS REDUCCION TRIGONOMETRICA
● 𝑠𝑒𝑛2𝐴+𝑐𝑜𝑠2𝐴 = 1
● 𝑠𝑒𝑛2𝐴 = 1 − 𝑐𝑜𝑠2𝐴
● 𝑐𝑜𝑠2𝐴 = 1 − 𝑠𝑒𝑛2𝐴
● 𝑡𝑔2𝐴 = 𝑠𝑒𝑐2𝐴 − 1
● 𝑐𝑡𝑔2𝐴 = 𝑐𝑠𝑐2𝐴 − 1
● 𝑠𝑒𝑐2𝐴 = 𝑡𝑔2𝐴 + 1
● 𝑐𝑠𝑐2𝐴 = 𝑐𝑡𝑔2𝐴 + 1
FUNCIONES DE ANGULOS NOBLES
● 𝑠𝑒𝑛 2𝑥 = 2 𝑠𝑒𝑛 𝑥 cos 𝑥
● 𝑐𝑜𝑠 2𝑥 = 𝑐𝑜𝑠2𝑥−𝑠𝑒𝑛2𝑥
● 𝑡𝑔 2𝑥 = 2 𝑡𝑔 𝑥
1−𝑡𝑔2𝑥
FUNCIONES DE SUMA Y DIFERENCIA DE DOS ANGULOS
● 𝑠𝑒𝑛 (𝑥 +𝑦)= 𝑠𝑒𝑛 𝑥 cos 𝑦 + cos 𝑥 𝑠𝑒𝑛 𝑦
● 𝑠𝑒𝑛 (𝑥 −𝑦)= 𝑠𝑒𝑛 𝑥 cos 𝑦 − cos 𝑥 𝑠𝑒𝑛 𝑦
● 𝑐𝑜𝑠 (𝑥 +𝑦)= 𝑐𝑜𝑠 𝑥 cos 𝑦 − sen 𝑥 𝑠𝑒𝑛 𝑦
● 𝑐𝑜𝑠 (𝑥 −𝑦)= 𝑐𝑜𝑠 𝑥 cos 𝑦 + sen 𝑥 𝑠𝑒𝑛 𝑦
● 𝑡𝑔 (𝑥+𝑦)=𝑡𝑔 𝑥 + 𝑡𝑔 𝑦
1−𝑡𝑔 𝑥 𝑡𝑔 𝑦
INTEGRALES ORDINARIAS 18-23
18) ∫𝑑𝑣
𝑣2+𝑎2=1
𝑎 𝑎𝑟𝑐𝑡𝑔 𝑣
𝑎+ 𝐶
19.a) ∫𝑑𝑣
𝑣2−𝑎2=1
2𝑎 𝑙𝑛 ( 𝑣−𝑎
𝑣+𝑎 ) + 𝐶
19.b) ∫𝑑𝑣
𝑎2−𝑣2=1
2𝑎 𝑙𝑛 ( 𝑎+𝑣
𝑎−𝑣 ) + 𝐶
20) ∫𝑑𝑣
√𝑎2−𝑣2= 𝑎𝑟𝑐𝑠𝑒𝑛 𝑣
𝑎+ 𝐶
21) ∫𝑑𝑣
√𝑣2±𝑎2=ln(𝑣+√𝑣2±𝑎2 ) + 𝐶
22) ∫√𝑎2−𝑣2𝑑𝑣 =𝑣
2√𝑎2−𝑣2+𝑎2
2𝑎𝑟𝑐𝑠𝑒𝑛𝑣
𝑎+ 𝐶
23) ∫√𝑣2±𝑎2𝑑𝑣 =𝑣
2√𝑣2±𝑎2±𝑎2
2ln ( 𝑣 + √𝑣2± 𝑎2 ) + 𝐶
DERIVADA FUNCION INVERSA
● 𝑑
𝑑𝑣 (arc sen v) = 𝑑
𝑑𝑣(𝑣)
√1−𝑣2
● 𝑑
𝑑𝑣 (arc cos v) = − 𝑑
𝑑𝑣(𝑣)
√1−𝑣2
● 𝑑
𝑑𝑣 (arc tg v) = 𝑑
𝑑𝑣(𝑣)
1+𝑣2
● 𝑑
𝑑𝑣 (arc ctg v) = − 𝑑
𝑑𝑣(𝑣)
1+𝑣2
● 𝑑
𝑑𝑣 (arc sec v) = 𝑑
𝑑𝑣(𝑣)
𝑣√𝑣2−1
● 𝑑
𝑑𝑣 (arc csc v) = − 𝑑
𝑑𝑣(𝑣)
𝑣√𝑣2−1
INTEGRAL TRIGONOMETRICA
CASO 1
∫𝑠𝑒𝑛 𝑚𝑢∗𝑐𝑜𝑠 𝑚𝑢∗𝑑𝑢 ; queda ∫𝑉 𝑛 ; m/n (positivo impar)
CASO 2
∫𝑡𝑔 𝑛𝑢∗𝑑𝑢 o ∫𝑐𝑡𝑔 𝑛𝑢∗𝑑𝑢; donde n (número entero +/ - / 0)
CASO 3
∫𝑠𝑒𝑐 𝑛𝑢 ∗𝑑𝑢 o ∫𝑐𝑠𝑐 𝑛𝑢 ∗ 𝑑𝑢; donde n (entero positivo par)
CASO 4
∫𝑡𝑔 𝑚𝑢 ∗ 𝑠𝑒𝑐 𝑛𝑢 ∗ 𝑑𝑢 o ∫𝑐𝑡𝑔 𝑚𝑢 ∗ 𝑐𝑠𝑐 𝑛𝑢∗𝑑𝑢;
donde n (entero positivo par)
CASO 5
1) 𝑆𝑒𝑛 𝑥 𝐶𝑜𝑠 𝑥 = 1
2𝑆𝑒𝑛 2𝑥
2) 𝑆𝑒𝑛2 𝑥 = 1
2−1
2𝐶𝑜𝑠 2𝑥
3) 𝐶𝑜𝑠2 𝑥 = 1
2+1
2𝐶𝑜𝑠 2𝑥
CASO 6
1) 𝑆𝑒𝑛 𝑚𝑥 𝐶𝑜𝑠 𝑛𝑥 𝑑𝑥 =−cos(𝑚+𝑛)𝑥
2(𝑚+𝑛) −cos(𝑚−𝑛)𝑥
2(𝑚−𝑛)+𝐶
2) 𝑆𝑒𝑛 𝑚𝑥 𝑆𝑒𝑛 𝑛𝑥 𝑑𝑥 =−sen(𝑚+𝑛)𝑥
2(𝑚+𝑛) +sen(𝑚−𝑛)𝑥
2(𝑚−𝑛)+𝐶
3) 𝐶𝑜𝑠 𝑚𝑥 𝐶𝑜𝑠 𝑛𝑥 𝑑𝑥 =sen(𝑚+𝑛)𝑥
2(𝑚+𝑛) +sen(𝑚−𝑛)𝑥
2(𝑚−𝑛)+𝐶
SUSTITUCION TRIGONOMETRICA
1) √𝑎2−𝑢2→ 𝑢2= 𝑎2𝑠𝑒𝑛2𝑍
2) √𝑢2+𝑎2→ 𝑢2= 𝑎2𝑡𝑔2𝑍
3) √𝑢2−𝑎2→ 𝑢2= 𝑎2𝑠𝑒𝑐2𝑍
INTEGRACION POR PARTES
∫ 𝑢 𝑑𝑣 =𝑢∗𝑣 − ∫𝑉∗𝑑𝑈
