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# Standard Differentials and Integrals

## Equations for differentials and integrals.

### Differentials

(d/dx) xn = n xn-1                     (1)

(d/dx) ln x = 1 / x                     (2)

(d/dx) eax = a eax                     (3)

(d/dx) ax = ax ln a                     (4)

(d/dx) xx = xx(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)

∫ eax dx = eax / a      a ≠ 0                    (16)

∫ ax dx = ax / 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) dx = tan-1 x      |x| < 1                          (26)

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## Citation

• The Engineering ToolBox (2011). Standard Differentials and Integrals. [online] Available at: https://www.engineeringtoolbox.com/differentials-integrals-d_1799.html [Accessed Day Month Year].

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9.19.12