Tabla de derivadas e integrales

Tabla de Derivadas e Integrales
Dr. Juli´n Gpe. Tapia Aguilar a juliangpe@yahoo.com.mx E S F M del I P N Enero de 2008

1.
1.1.

Derivadas
Logaritmos y Exponenciales
1 · Dx [u] u 1 Dx [ logb u] = · Dx [u] (ln b) · u Dx [eu ] = eu · Dx [u] Dx [ ln u] = Dx [a ] = (ln a) · a · Dx [u]
u u

(1) (2) (3) (4)

1.2.

Funciones Trigonom´tricas e

Dx [ sin u] = cos u · Dx [u] Dx [ cos u] =− sin u · Dx [u] Dx [ tan u] = sec u · Dx [u] Dx [ cot u] = − csc u · Dx [u] Dx [ sec u] = sec u tan u · Dx [u] Dx [ csc u] = − csc u cot u · Dx [u] 1 Dx [ sin−1 u] = √ · Dx [u] 1 − u2 1 · Dx [u] Dx [ cos−1 u] = − √ 1 − u2 1 Dx [ tan−1 u] = · Dx [u] 1 + u2 1 · Dx [u] Dx [ cot−1 u] = − 1 + u2 1
2 2

(5) (6) (7) (8) (9) (10) (11) (12) (13) (14)

C´lculo II – Juli´n Gpe. Tapia Aguilar a a 1 √ ·Dx [u] u u2 − 1 1 · Dx [u] Dx [ csc−1 u] = − √ u u2 − 1 Dx [ sec−1 u] =

2

(15) (16)

1.3.

Funciones Hiperb´licas o

Dx [ sinh u] = cosh u · Dx [u] Dx [ cosh u] = sinh u · Dx [u] Dx [ tanh u] = sech u · Dx [u] Dx [ coth u] = −csch u · Dx [u] Dx [sechu] = −sechu tanh u · Dx [u] Dx [csch u] = −csch u coth u · Dx [u] 1 · Dx [u] Dx [ sinh−1 u] = √ 2+1 u 1 Dx [ cosh−1 u] = √ · Dx [u] 2−1 u 1Dx [ tanh−1 u] = · Dx [u] 1 − u2 1 Dx [ coth−1 u] = · Dx [u] 1 − u2 1 Dx [sech−1 u] = − √ · Dx [u] u 1 − u2 1 Dx [csch −1 u] = − √ · Dx [u] u 1 + u2
2 2

(17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28)

2.
2.1.

Integrales
Logaritmos y Exponenciales
ax = ex·ln a , loga x = ln x ln a (29) x2 4 x3 9 (30) (31) Recuerde que:

ln x dx = x ln x − x x · ln x dx = x2 · ln x dx =x2 ln x − 2 x3 ln x − 3

C´lculo II – Juli´n Gpe. Tapia Aguilar a a xn+1 xn+1 ln x − n+1 (n + 1)2 1 ax ·e a 1 · bax a ln b x 1 ( − 2 ) · eax a a 2 x2 2x ( − 2 + 3 ) · eax a a a xn ax n e − xn−1 · eax dx. a a

3

xn · ln x dx = eax dx = bax dx = x · eax dx = x2 · eax dx = xn · eax dx =

(32) (33) (34) (35) (36) (37)

2.2.

Funciones Trigonom´tricas e
1 sin ax dx = − cos ax. a 1 sin ax.cos ax dx = a 1 1 tan ax dx = − ln | cos ax| = ln | sec ax|. a a 1 1 cot ax dx = ln | sin ax| = − ln | csc ax|. a a 1 1 sec ax dx = ln | sec ax + tan ax| = − ln | sec ax − tan ax|. a a 1 1 csc ax dx = − ln | csc ax + cot ax| = ln | csc ax − cot ax|. a a 1 sec2 ax dx = tan ax. a 1 csc2 ax dx = − cot ax. a 1 sec ax tan x dx = sec ax. a 1 csc ax cot x dx = − csc ax. a 1 x x √ dx = sin−1 ( ) = −cos−1 ( ) + constante. 2 − x2 a a a 1 x 1 x 1 dx = tan−1 ( ) = − cot−1 ( ) + constante. 2 + x2 a a a a a

(38) (39) (40) (41) (42) (43) (44) (45) (46) (47) (48) (49)

C´lculo II – Juli´n Gpe. Tapia Aguilar a a 1 √ dx = 2 − a2 x x 1 x 1 x sec−1 ( ) = − csc−1 ( ) + constante. a a a a

4

(50)

Para algunas potencias de senos y cosenos es importante recordar las identidades, conocidas como del´ngulo doble: a sen(2x) = 2 sin x cos x, 1 sen2 x = (1 − cos 2x), 2 1 cos2 x = (1 + cos 2x). 2 (51) (52) (53) (54) (55) (56) (57) (58) (59) n = 1. (60)

sin2 x dx = sin3 x dx = sin4 x dx = cos2 x dx = cos3 x dx = cos4 x dx =

1 1 x − sen(2x). 2 4 cos3 x 3 cos x cos(3x) − cos x + =− + . 3 4 12 3x sen(2x) sen(4x) − + . 8 4 32 1 1 x + sen(2x). 2 4 3 sen x sen(3x) sen3 x = + . sen x − 3 4 12 3xsen(2x) sen(4x) + + . 8 4 32

tan2 x dx = tan x − x. tan3 x dx = tan4 x dx = tann x dx = tan2 x + ln | cos x|. 2 tan3 x − tan x + x. 3 tann−1 x − tann−2 x dx, n−1

Para productos de senos y/o cosenos con argumentos diferentes, las identidades importantes son, sen(α ± β) = sen α cos β ± sen β cos α, cos(α ± β) = cos α cos β sin β sin α,

que con simples manipulaciones de sumas y restas nos danlas siguientes identidades: sen α cos β = sen α sen β = cos α cos β = 1 [sen(α + β) + sen(α − β)] 2 1 [cos(α − β) − cos(α + β)] 2 1 [cos(α + β) + cos(α − β)] 2

C´lculo II – Juli´n Gpe. Tapia Aguilar a a Algunas f´rmulas de integraci´n (para α = ±β) son: o o sen(αx) cos(βx) dx = sen(αx) sen(βx) dx = cos(αx) cos(βx) dx = 1 cos[(α + β)x] cos[(α − β)x] − − 2 α+β α−β 1 sen[(α − β)x] sen[(α +...