Tavola sviluppi di Taylor

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TA VOLA DEGLI SVILUPPI DI TAYLOR DELLE FUNZIONI ELEMENTARI PER x → 0. ex = 1 + x + x2 2 + x3 6 + · · ·+ xn n! + o(xn) sin x = x − x3 6 + x5 5! + · · ·+ (−1)n (2n + 1)! x2n+1 + o(x2n+2) cos x = 1 − x2 2 + x4 4! + · · ·+ (−1)n (2n)! x2n + o(x2n+1) tan x = x + x3 3 + 2 15 x5 + 17 315 x7 + 62 2835 x9 + o(x10) sinh x = x + x3 6 + x5 5! + · · ·+ x2n+1 (2n + 1)! + o(x2n+2) cosh x = 1 + x2 2 + x4 4! + · · ·+ x2n (2n)! + o(x2n+1) tanh x = x − x3 3 + 2 15 x5 − 17 315 x7 + 62 2835 x9 + o(x10) 1 1 − x = 1 + x + x2 + x3 + · · ·+ xn + o(xn) log(1 + x) = x − x2 2 + x3 3 + · · ·+ (−1)n+1 n xn + o(xn) arctan x = x − x3 3 + x5 5 + · · ·+ (−1)n 2n + 1 x2n+1 + o(x2n+2) arctanh x = x + x3 3 + x5 5 + · · ·+ x2n+1 2n + 1 + o(x2n+2) (1 + x)α = 1 + αx + α(α − 1) 2 x2 + α(α − 1)(α − 2) 6 x3 + · · ·+ ( α n ) xn + o(xn) con ( α n ) = α(α − 1)(α − 2) · · · · ·(α − n + 1) n! 1 TA VOLA DI PRIMITIVE DI FUNZIONI ELEMENTARI ∫ xa dx = xa+1 a + 1 + C se a ̸= −1 ∫ 1 x dx = log |x|+ C ∫ ax dx = ax log a + C ∫ cos x dx = sin x + C ∫ sin x dx = − cos x + C ∫ 1 cos2 x dx = tan x + C ∫ 1 sin2 x dx = − cotan x + C ∫ cosh x dx = sinh x + C ∫ sinh x dx = cosh x + C ∫ 1 cosh2 x dx = tanh x + C ∫ 1 sinh2 x dx = − cotanh x + C ∫ 1 a2 + x2 dx = 1 a arctan x a + C ∫ 1√ 1 − x2 dx = arcsin x + C ∫ 1√ x2 + 1 dx = arcsinh x + C = log ( x + √ x2 + 1 ) + C ∫ 1√ x2 − 1 dx = arccosh x + C = log ( x + √ x2 − 1 ) + C per x > 1 2