Formulario, Istituzioni di matematica

Formulario: tavola delle derivate fondamentali

Regole di derivazione

Formulario: logaritmi

log_ab = x ⇔ a^x = b, a > 0, a ≠ 1, b > 0, ∀x ∈ R

log_bc = log_ac / log_ba, a > 0, a ≠ 1, b > 0, b ≠ 1, c > 0

Proprietà:

• log_a(m · n) = log_am + log_an, a > 0, a ≠ 1, m > 0, n > 0
• log_a(mⁿ / n^m) = log_am - log_an, a > 0, a ≠ 1, m > 0, n > 0
• log_amⁿ = n · log_am, a > 0, a ≠ 1, m > 0, n ∈ R
• log_aⁿ√m = log_am^1/n = 1/n log_am, a > 0, a ≠ 1, m > 0, n ∈ N₀

Formulario: limiti ricorrenti

• lim_x→0 sin x/x = 1
• lim_x→0 tg x/x = 1
• lim_x→0 (1 - cos x)/x = 0
• lim_x→0 (1 - cos x)/x² = 1/2
• lim_x→0 arcsin x/x = 1
• lim_x→0 arctg x/x = 1
• lim_x→1 (arccos x)²/(1-x) = 2
• lim_x→0 (e^x-1)/log_e e = 1
• lim_x→∞ (1 + 1/x)^x = e
• lim_x→0 (1 + x)^1/x = e
• lim_x→∞ log_a (1 + 1/x)^x = log_a e
• lim_x→∞ log_e (1 + 1/x)^x = log_e e = 1
• lim_x→0 log_e (1 + x)/x = 1
• lim_x→0 x/log_a (1+x) = 1/log_a e
• lim_x→0 x/log_e (1+x) = 1/log_e e = 1
• lim_x→0 a^x-1/x = 1/log_a e

Formulario: tavola degli integrali indefiniti

Integrali indefiniti fondamentali

• ∫f'(x)dx = f(x) + c
• ∫a dx = ax + c
• ∫xⁿ dx = xⁿ⁺¹ / (n+1) + c; con n ≠ -1
• ∫x^-1 dx = log |x| + c
• ∫sin x dx = -cos x + c
• ∫cos x dx = sin x + c
• ∫(1 + tg² x) dx = ∫1 / cos² x dx = tg x + c
• ∫(1 + ctg² x) dx = ∫1 / sin² x dx = -ctg x + c
• ∫Sh x dx = Ch x + c
• ∫Ch x dx = Sh x + c
• ∫e^x dx = e^x + c
• ∫e^kx dx = e^kx / k + c
• ∫a^x dx = a^x / log_e a + c

Integrali notevoli

• ∫1 / sin x dx = log |tg (x/2)| + c
• ∫1 / cos x dx = log |tg (x/2 + π/4)| + c
• ∫1 / √(1 - x²) dx = {arcsin x + c; -arccos x + c}
• ∫-1 / √(1 - x²) dx = {arccos x + c; -arcsin x + c}
• ∫1 / (1 + x²) dx = arctg x + c
• ∫1 / (x² - 1) dx = 1/2 log |(1 + x) / (1 - x)| + c
• ∫1 / √(x² - 1) dx = log |x + √(x² - 1)| + c
• ∫1 / √(1 + x²) dx = {arcsin x + c; log |x + √(1 + x²)| + c}
• ∫1 / √(x² + a²) dx = log |x + √(x² + a²)| + c
• ∫√(x² + a²) dx = x/2 √(x² + a²) + a² / 2 log [x + √(x² + a²)] + c
• ∫√(a² - x²) dx = 1/2 {a² arcsin x/a + x √(a² - x²)} + c