Integrales

sin2 ax cos2 bx dx =

x sin 2ax sin[2(a − b)x] − − 4 8a 16(a − b) sin 2bx sin[2(a + b)x] − (79) + 8b 16(a + b) x sin 4ax − 8 32a (80) (81) (82) (83)

xn cos ax dx = xn sin ax n − a a x sin ax dx = − x2 sin ax dx = xn sin ax dx = − xn a cos ax + n a xn−1 cos ax dx, n > 0 (101) 2− a3 xn−1 sin ax dx x cos ax sin ax + a a2 cos ax + 2x sin ax a2 (98)

Tabla de Integrales
neoparaiso.com/imprimirFormas B´sicas a
(99) c f (x) dx = c f (x) dx (1) x2 x dx = − a + bx2 b

sin2 ax cos2 ax dx =

√

a arctan b3/2

√ bx √ a

(16) (17) (18)

1 tan ax dx = − ln cos ax a 1 tan ax dx = −x + tan ax a
2

a2 x2

(100)

a x2 x3 − dx = ln |a + bx2 | 2 a + bx 2b 2b2 x a + ln |a + x| dx = (x + a)2 a+x 1 a+x 1 dx = ln , a=b (x + a)(x + b) b−a b+x

a f (x) + b g(x) dx = a

f (x) dx+ b

g(x) dx (2)

1 1 sec2 ax tan3 ax dx = ln cos ax + a 2a tann ax dx = 1 tann−1 ax − a(n − 1) tann−2 ax dx, n = 1

axn dx =

Productos de Funciones Trigonom´tricas y Exponencial e
(84) 1 ex sin x dx = ex (sin x − cos x) 2 ebx sin ax dx = (102)

a xn+1 , n = −1 n+1 f (x)n+1 , n = −1 n+1

(3)

(19)

af (x)n · f ′ (x) dx =

(4)

1 2 2ax + b dx = √ arctan √ ax2 + bx + c 4ac − b24ac − b2 (20) x 1 dx = ln |ax2 + bx + c| ax2 + bx + c 2a 2ax + b b arctan √ − √ 4ac − b2 a 4ac − b2 dx = (ax2 + bx + c)n 2ax + b (n − 1)(4ac − b2 )(ax2 + bx + c)n−1 dx (2n − 3)2a + (n − 1)(4ac − b2 ) (ax2 + bx + c)n−1 x dx = (ax2 + bx + c)n bx + 2c (n − 1)(4ac − b2 )(ax2 + bx + c)n−1 b(2n − 3) dx − (n − 1)(4ac − b2 ) (ax2 + bx + c)n−1

sec x dx = ln | sec x + tan x| sec2 ax dx = sec3 x dx = 1tan ax a

(85) (86)

1 ebx (b sin ax − a cos ax) (103) a2 + b2 (104)

a dx = a ln x x 1 1 dx = ln |ax + b| ax + b a f ′ (x) dx = ln |f (x)| f (x) bax dx = bax a ln b

(5) (6) (7) (8)

1 ex cos x dx = ex (sin x + cos x) 2
bx

(21)

1 1 sec x tan x + ln | sec x + tan x| (87) 2 2 sec x tan x dx = sec x sec2 x tan x dx = 1 sec2 x 2 (88) (89) (90) (91) (92)

1 e cos ax dx = 2 ebx (a sinax + b cos ax) (105) a + b2 1 xe sin x dx = ex (cos x − x cos x + x sin x) (106) 2
x

(22)

secn x tan x dx =

1 secn x, n = 0 n

xex cos x dx =

1 x e (x cos x − sin x + x sin x) (107) 2

u dv = uv −

v du = u

dv −

v du

(9)

x = ln | csc x − cot x| csc x dx = ln tan 2 1 csc2 ax dx = − cot ax a

Integrales con Funciones Trigonom´tricas Inversas e
arcsin x dx = xarcsin x + 1 − x2 1− x2 (108)

Integrales de Funciones Racionales
1 1 dx = − (x + a)2 x+a (x + a)n dx = (109) (x + a)n+1 , n = −1 n+1 (10)

(23)

1 1 csc3 x dx = − cot x csc x + ln | csc x − cot x| (93) 2 2 1 cscn x cot x dx = − cscn x, n = 0 n sec x csc x dx = ln | tan x| (94) (95)

(11)

Integrales con Ra´ ıces
√ ax + b dx = 2 (ax + b)3/2 3a (24) (25) (26)

arccos x dx = x arccos x −1 arctan x dx = x arctan x − ln 1 + x2 2 arccot x dx = x arccot x + 1 ln 1 + x2 2

(110)

(x + a)n+1 ((n + 1)x − a) x(x + a)n dx = (n + 1)(n + 2) 1 dx = arctan x 1 + x2 1 1 dx = √ arctan x a + bx2 ab 1 x dx = ln |a + bx2 | a + bx2 2b b a

(12) √

1 2√ √ ax + b dx = a ax + b ax + b dx = 2b 2x + 3a 3 √ ax + b

Productos de Funciones Trigonom´tricas y Monomios e
x cos ax dx = x 1 cos ax +sin ax a2 a (96)

(111)

(13)

arcsec x dx = x arcsec x − arcosh x arccsc x dx = x arccsc x + arcosh x

(112)

(14) (15)

√ √ 2 ax + b(3a2 x2 + abx − 2b2 ) (27) x ax + b dx = 15a2 (ax + b)3/2 dx = 2 (ax + b)5/2 5a (28)

(113)

2x cos ax a2 x2 − 2 x2 cos ax dx = + sin ax a2 a3

(97)

√

√ x 2 dx = 2 (ax ∓ 2b) ax ± b 3a ax ± b x dx = − a−x x(a − x) x(a − x) x−a

(29)

x √dx = − a2 − x2 √

a2 − x2

(41)

ln ax2 + bx + c dx = − 2x +

1 a

− a arctan

(30)

x2 1 dx = x x2 ± a2 2 ± a2 2 x 1 ∓ a2 ln x + x2 ± a2 2

b + x ln ax2 + bx + c 2a

2ax + b 4ac − b2 arctan √ 4ac − b2 (52)

Integrales con Funciones Trigonom´tricas e
1 sin ax dx = − cos ax a sin2 ax dx = x sin 2ax − 2 4a (66) (67)

(42) x ln(ax + b) dx = bx 1 2 − x 2a 4 1 b2 + x2 − 2 2 a...