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IDENTIDADES TRIGONOMETRICAS (Trigonometría Plana) FORMULAS DERIVACIÓN FORMULAS DIFERENCIALES FORMULAS INTEGRACION

Sen X = 1

SenX + SenY = 2Sen ½ (X+Y)Cos ½ (X-Y) d x (c) = 0dx

d x (x) = 1dx dc = 0 ∫ dx = x + CCsc X

Cos X =1

SenX - SenY = 2Cos ½ (X+Y)Sen ½ (X-Y) d a (u + v - w) = d s (u) + d x (v) – d X (w)dx dx dx dx d (u + v - w) = du + dv - dw ∫ C (du) = C ∫ duSec X

Tg X = 1

Cos + CosY = 2Cos ½ (X+Y) Cos ½ (X–Y) d 0 (c u) = c d (u)dx dx d (c u) = c du ∫ ( du + dv – dw) = ∫ du + ∫ dv - ∫ dw

Ctg X

Ctg X =1

CosX – CosY = 2Sen ½ (X+Y) Sen ½ (X–Y) d (u v) = u d (v) + v d (u)dx dx dx d (u v) = u dv + v du

∫ un du = u n+1 + C n + 1Tg X

Sec X = 1 SenX + SenY =Tg ½ (X+Y) d_ u

dx v v d (u) - u d _(v) dx dx q d u

v = v du - u dv ∫ u-1 du = ∫ du = Ln U + C u Cos X SenX - SenY Tg ½ (X-Y) v2 v2

Csc X =1

Sen (X+Y) Sen (X-Y) = Sen2X – Sen2Y d un = nun-1 d (u)dx dx

dun = nun-1 du∫au du = a u + C ; a > 0 ; a ≠ 1 Ln aSen X

Sen X = Tg X Sen (X+Y) Sen (X-Y) = Cos2Y – Cos2X d Xn = nXn-1

dx d Sen u = Cos u d u ∫ eu du = eu +CCos XCos X = Ctg X Cos (X+Y) Cos (X-Y) = Cos2X – Sen2Y d u

dx c d (u) dx x c

d Cos u = - Sen u d u ∫ Sen u du = - Cos u + C Sen X

Sen 2 X + Cos 2 X = 1 Cos (X+Y) Cos (X-Y) = Cos2Y – Sen2X d_ c dx u

_ c d (u) u2 dx

d Tg u = Sec2 u d u ∫ Cos u du = Sen u + C

Sen 2 X = 1 - Cos 2 X Funciones Trigonometricas

d Ctg u = - Csc2 u d u ∫ Tg u du = Ln / Sec u / + C = - Ln Cos uFUNCIONES DE 2X

Cos 2 X = 1 - Sen 2 X Sen 2X = 2 SenX CosX d_ Sen u = Cos u d (u)dx dx

d Sec u = Sec u Tg u d u ∫ Ctg u du = Ln / Sen / + C

Sec 2 X = 1 + Tg 2 X Cos 2X = Cos2X – Sen2X d_ Cos u = - Sen u d (u)dx dx

d Csc u = - Csc u Ctg u d u ∫ Sec u du = Ln / Sec u + Tg u / + C

Tg 2 X = Sec 2 X - 1 Cos 2X = 2Cos2 X – 1 d_ Tg u = Sec2 u d (u)dx dx

d ArcSen u = du

∫ Csc u du = Ln / Csc u – Ctg u / + C

Csc 2 X = 1 + Ctg 2 X Cos 2X = 1 – 2Sen2 X d Ctg u = - Csc2 u d (u)dx dx

d ArcCos u = _ du x

∫ sec2 u du = Tg u + C

Ctg 2 X = Csc 2 X – 1 Tg 2X =2 TgX

d Sec u = Sec u Tg u d (u)dx dx

d ArcTg u = du

∫ Csc2 u du = - Ctg u + C1 – Tg2X 1 + u2

Sen (X+Y) = SenX CosY + CosX SenY Ctg 2X =Ctg2 X – 1 d Csc u = - Csc u Ctg u d (u)

dx dx d ArcCtg u = _ du c ∫ Sec u Tg u du Sec u + C

2 Ctg X 1 + u2

Cos (X+Y) = CosX CosY – SenX SenYFUNCIONES DE ½ X Trigonometricas Inversas

d ArcSec u = du

∫ Csc u Ctg u du = - Csc u + C

Sen (X-Y) = SenX CosY – CosX SenY Sen2 ½ X =1 – Cos X

d ArcSen u = dx

1 1 1 d (u)dx

d ArcCsc u = _ du f ∫ du = ArcSen u + C

a2

Cos (X-Y) = CosX CosY + SenX SenY Cos2 ½ X =1 + Cos X

d ArcCos u = dx

_ 1 1 d (u)dx

d loga u = log a e du u

∫ du = 1 ArcTg u + C a2 + u2 a a2

Tg (X+Y) =TgX + TgY

Tg ½ X =1 – Cos X d ArcTg u =

dx1 1 1 d (u)

dx d ln u = du u

∫ du = 1 ArcSec u + C

a a1- TgX TgY Sen X 1 + u2

Ctg (X+Y) =CtgX CtgY – 1

Ctg ½ X =1 + Cos X d ArcCtg u= -

dx1 d (u)

dx d au = au ln a du∫ du = 1 Ln / u – a / + C u2 – a2 2a u + aCtgY + CtgX Sen X 1 + u2

Tg (X-Y) =TgX – TgY

FUNCIONES DE 3Xd ArcSec u = dx

1 d (u)dx d eu = eu du

∫ du = 1 Ln / a + u / + C a2 – u2 2a a – u 1 + TgX TgY

Ctg(X-Y) =CtgX CtgY + 1

Sen 3X = 3SenX – 4Sen3X d ArcCsc u = dx

_ 1 1 d (u)dx

∫ du = Ln ( u + ) + C

CtgY – Ctg X

Cos 3X = 4Cos3X – 3CosXLogaritmicas Exponenciales

∫ du = Ln ( u + ) + C

Tg 3X = 3TgX – Tg3X d Log a u = Log a e d u

dx u dx ∫ du = ½ u + ½ a2 ArcSen u/a

1 – 3Tg3X

ING. MARIO A. BARRERA MORENO

d Ln u = 1 d udx u dx

∫ du = ½ u + ½ a2 Ln(u+

)

d au = au Ln a d udx dx

∫ du = ½ u - ½ a2 Ln( u +

)

d eu = eu d udx dx

d uv = uv ln u d v + vuv-1 d udx dx dx