Differentiation & Integration formulas PDF

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Differentiation Formulas The following table provides the differentiation formulas for common functions. The first six rows correspond to general rules (such as the addition rule or the product rule) whereas the remaining rows contain the formulas for specific functions.

F (x)

F ′ (x)

Addition

f (x) ± g (x)

f ′ (x) ± g ′ (x)

Linearity

af (x)

af ′ (x)

Product Rule

f (x)g (x)

f ′ (x)g (x) + f (x)g ′ (x)

Quotient Rule

f (x) g(x)

f ′ (x)g(x)−f (x)g ′ (x) (g (x))2

Chain Rule

f (g (x))

f ′ (g (x)) · g ′ (x) 1 f ′ (f −1 (x))

f −1 (x) Basic functions

xn

for any real n

ex ax

Trig functions

nxn−1 ex

(a > 0)

(ln a)ax

ln x

1 x

sin x

cos x

cos x

− sin x

tan x arctan x = tan−1 x arcsin x = sin−1 x

1 = cos2 x 1 1+x2 √ 1 1−x2

Hyperbolic Trig sinh x

cosh x

cosh x

sinh x

tanh x

1 cosh2 x √ 1 1+x2

sinh−1 x tanh−1 x

1 1−x2

1 + tan2 x

Integration Formulas The following list provides some of the rules for finding integrals and a few of the common antiderivatives of functions. Linearity Substitution Integration by parts







af (x) + bg (x) dx = a f (x) dx + b g (x) dx  f (w(x))w′ (x) dx = f (w) dw   u(x)v′ (x) dx = u(x)v(x) − u′ (x)v(x) dx



Basic Functions 

1 dx = ln |x| + C x  ax ax dx = +C ln a

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xn+1 +C n+1  1 eax dx = ex + C a xn dx =

Trigonometric functions 

sin x dx = − cos x + C



cos x dx = sin x + C

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1 dx = tan x + C cos2 x

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tan x dx = − ln | cos x| + C

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cot x dx = ln | sin x| + C

sinh x dx = cosh x + C



cosh x dx = sinh x + C

tanh x dx = ln(cosh x) + C



coth x dx = ln | sinh x| + C

Hyperbolic Trig functions  

Functions with a2 ± x2 

 x dx √ +C = sin−1 a a2 − x2    dx 1  x + a ln = +C 2 2 x−a 2a a −x   x dx √ +C = cosh−1 2 2 a x −a



  dx 1 −1 x +C = tan a2 + x2 a a



 x dx √ +C = sinh−1 2 2 a x +a



arcsin x dx = x arcsin x +

Inverse Functions 

ln x dx = x ln x − x + C



arctan x = x arctan x −

1 ln(1 + x2 ) + C 2



1 − x2 + C...