Chapter 4 | Applications of Derivatives
489
Differentiation Formula Indefinite Integral
d dx (
∫ kdx = ∫ kx 0 dx = kx + C
k ) =0
d dx (
n +1 n +1 +
x n ) = nx n −1
∫
x n dn = x
C for n ≠−1
d dx (ln|
∫ 1
x |) = 1 x
x dx = ln| x | + C
d dx (
∫ e x dx = e x + C
e x ) = e x
d dx (sin
∫ cos xdx = sin x + C
x ) =cos x
d dx (cos
∫ sin xdx =−cos x + C
x ) =−sin x
d dx (tan
∫ sec 2 xdx = tan x + C
x ) = sec 2 x
d dx (csc
∫ csc x cot xdx =−csc x + C
x ) =−csc x cot x
d dx (sec
∫ sec x tan xdx = sec x + C
x ) = sec x tan x
d dx (cot
∫ csc 2 xdx =−cot x + C
x ) =−csc 2 x
⎛ ⎝ sin −1 x
⎞ ⎠ = 1
d dx
∫ 1
= sin −1 x + C
1− x 2
1− x 2
⎛ ⎝ tan −1 x
⎞ ⎠ = 1
d dx
∫
1 1+ x 2
dx = tan −1 x + C
1+ x 2
⎛ ⎝ sec −1 | x | ⎞
d dx
∫
1 x x 2 −1
dx = sec −1 | x | + C
⎠ = 1
x x 2 −1
Table 4.13 Integration Formulas
From the definition of indefinite integral of f , we know
∫ f ( x ) dx = F ( x )+ C
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