Standard Differentials and Integrals

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Differentials

(d/dx) xⁿ = n x^n-1    (1)

(d/dx) ln x = 1 / x     (2)

(d/dx) e^ax = a e^ax    (3)

(d/dx) a^x = a^x ln a    (4)

(d/dx) x^x = x^x(1 + ln x)   (5)

(d/dx) sin x = cos x   (6)

(d/dx) cos x = - sin x     (7)

(d/dx) tanx = sec² x    (8)

(d/dx) cot x = - cosec² x    (9)

(d/dx) sin^-1x = 1 / (1 - x²)^1/2   (10)

(d/dx) cos^-1x = -1 / (1 - x²)^1/2    (11)

(d/dx) tan^-1x = 1 / (1 + x²)      (12)

(d/dx) cot^-1x = -1 / (1 + x²)      (13)

Integrals

∫ xⁿ dx = xⁿ⁺¹ / (n + 1)      n ≠ 1       (14)

∫ 1 / x dx = ln x       (15)

∫ e^ax dx = e^ax / a      a ≠ 0       (16)

∫ a^x dx = a^x / ln a     a > 0, a ≠ 1       (17)

∫ ln x dx = x (ln x - 1)       (18)

∫ sin x dx = - cos x       (19)

∫ cos x dx = sin x       (20)

∫ tan x dx = - ln(cos x)       (21)

∫ cot x dx = ln(sin x)      (22)

∫ sec² x dx = tan x       (23)

∫ cosec² x dx = - cot x       (24)

∫ 1 / (1 - x²)^1/2 dx = sin^-1 x      |x| < 1       (25)

∫ 1 / (1 + x²)^1/2 dx = tan^-1 x      |x| < 1       (26)

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