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Regras para derivadas As funções y, u, v, w são funções da variavel x e as letras k, a, n são consideradas constantes. A letra L representa o logaritmo neperiano. (1)y = k → y′ = 0 (2)y = x → y′ = 1 → (3)y = xn → y′ = n.xn−1 (4)y = k.xn → y′ = k.n.xn−1 → (5)y = un → y′ = n.un−1.u′ (6)y = u + v → y′ = u′ + v′ → (7)y = a.u → y′ = a.u′ (8)y = u.v → y′ = u′.v + u.v′ → (9)y = au → y′ = au.L(a).u′ (10)y = eu → y′ = eu.u′ → (11)y = logau → y′ = u′ u.logea (12)y = Lu → y′ = u ′ u → (13)y = uv → y′ = v.uv−1.u′ + uv.v′.L(u) (14)y = u v → y′ = vu ′ − uv′ v2 → (15)y = f(u);u = f(v)→ dy dv = dy du . du dv (16)x = f1(t), y = f2(t)→ dy dx = dy/dt dx/dt → (17)y = sen(u) → y′ = u′.cos(u) (18)y = cos(u) → y′ = −u′.sen(u) → (19)y = tg(u) → y′ = u′.sec2(u) (20)y = cotg(u) → y′ = −u′.cossec2(u) → (21)y = sec(u) → y′ = u′.sec(u)tg(u) (22)y = cossec(u) → y′ = −u′.cossec(u)cotg(u) → (23)y = arc sen(u) → y′ = u ′ √ 1− u2 (24)y = arc cos(u) → y′ = −u ′ √ 1− u2 → (25)y = arc tg(u) → y′ = u ′ 1 + u2 (26)y = arc cotg(u) → y′ = −u ′ 1 + u2 → (27)y = arc sec(u) → y′ = u ′ u(u2 − 1) (28)f ′(x) = g′(x) → f(x) = g(x) + k 1 Regras para Integrais As funções y, u, v, w são funções da variavel x e as letras k, a, n, c são consideradas constantes. (1) ∫ dx = x + c (2) ∫ xndx = xn+1 n + 1 + c, n 6= −1 (3) ∫ sen kx dx = −cos kx k + c (4) ∫ cos kx dx = sen kx k + c (5) ∫ sec2 x dx = tg k + c (6) ∫ cossec2 x dx = −cotg x + c (7) ∫ sec x tg x dx = sec x + c (8) ∫ cossec xcotg x dx = −cossec x + c (9) ∫ ekxdx = ekx k + c (10) ∫ 1 x dx = ln|x|+ c (11) ∫ dx√ 1− x2 = arc sen x + c (12) ∫ dx 1− x2 = arc tg x + c (13) ∫ dx x( √ x2 − 1) = arc sec x + c (14) ∫ axdx = ( 1 ln a )ax, a > 0, a 6= 1 (15) ∫ undu = un+1 n + 1 + c, n 6= −1 (16) ∫ 1 u du = ln|u|+ c (17) ∫ tg u du = −ln|cos u|+ c (18) ∫ cotg u du = ln|sen u|+ c (19) ∫ eudu = eu + c (20) ∫ audu = au ln|a| + c (21) ∫ sen(u)du = −cos(u) + c (22) ∫ cos(u)du = sen(u) + c (23) ∫ tg(u)du = ln|sec(u)|+ c (24) ∫ cotg(u)du = ln|sen(u)|+ c (25) ∫ sec(u)tg(u)du = sec(u) + c (26) ∫ cossec(u)cotg(u)du = −cossec(u) + c (27) ∫ sec(u)du = ln|sec(u) + tg(u)|+ c (28) ∫ cossec(u)du = ln|cossec(u)− cotg(u)|+ c (29) ∫ sec2(u)du = tg(u) + C (30) ∫ cossec2(u)du = cotg(u) + C (31) ∫ du√ a2 − u2 = arc sen( u a ) + c (32) ∫ du a2 + u2 = 1 a arc tg( u a ) + c (33) ∫ du√ u2 ± a2 = ln|u + √ u2 ± a2|+ c (34) ∫ du u2 − a2 = 1 2a ln|u− a u + a |+ c (35) ∫ du u( √ u2 − a2) = 1 a arc sec( u a ) (36) ∫ u dv = u.v − ∫ v.du 2
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