Calculus

                                                         Integrals 

Table of Integrals 

Power of x.

xn dx = x(n+1) / (n+1) + C  (n  -1)  Proof

1/x dx = ln|x| + C

Exponential / Logarithmic

ex dx = ex + C   Proof	bx dx = bx / ln(b) + C   Proof, Tip!
ln(x) dx = x ln(x) - x + C   Proof

Trigonometric

sin x dx = -cos x + C   Proof	csc x dx = - ln|CSC x + cot x| + C   Proof
COs x dx = sin x + C   Proof	sec x dx = ln|sec x + tan x| + C   Proof
tan x dx = -ln|COs x| + C   Proof	cot x dx = ln|sin x| + C   Proof

Trigonometric Result

COs x dx = sin x + C    Proof	CSC x cot x dx = - CSC x + C    Proof
sin x dx = COs x + C    Proof	sec x tan x dx = sec x + C    Proof
sec2 x dx = tan x + C    Proof	csc2 x dx = - cot x + C    Proof

Inverse Trigonometric

arcsin x dx = x arcsin x + (1-x2) + C

arccsc x dx = x arccos x - (1-x2) + C

arctan x dx = x arctan x - (1/2) ln(1+x2) + C

Inverse Trigonometric Result
 

dx  (1 - x2)  = arcsin x + C     dx  x (x2 - 1)  = arcsec|x| + C     dx  1 + x2  = arctan x + C

Useful Identitiesarccos x = /2 - arcsin x  (-1 <= x <= 1)  arccsc x = /2 - arcsec x  (|x| >= 1)  arccot x = /2 - arctan x  (for all x)

Hyperbolic

sinh x dx = cosh x + C    Proof	csch x dx = ln |tanh(x/2)| + C    Proof
cosh x dx = sinh x + C    Proof	sech x dx = arctan (sinh x) + C
tanh x dx = ln (cosh x) + C    Proof	coth x dx = ln |sinh x| + C   Proof

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Integral Identities

Formal Integral Definition:
 f(x) dx = lim _{(d -> 0)}  (k=1..n) f(X_(k)) (x_(k) - x_(k-1)) when...

a = x₀ < x₁ < x₂ < ... < x_n = b

d = max (x₁-x₀, x₂-x₁, ... , x_n - x_(n-1))

x_(k-1) <= X_(k) <= x_(k)     k = 1, 2, ... , n

 F '(x) dx = F(b) - F(a) (Fundamental Theorem for integrals of derivatives)

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a f(x) dx = a f(x) dx (if a is constant)

f(x) + g(x) dx = f(x) dx + g(x) dx

 f(x) dx = f(x) dx | (a b)

 f(x) dx +  f(x) dx =  f(x) dx

f(u) du/dx dx = f(u) du (integration by substitution)

Special Functions

Some of these functions I have seen defined under both intervals (0 to x) and (x to inf). In that case, both variant definitions are listed.
gamma = Euler's  constant = 0.5772156649...

(x) = Gamma(x) = t^{^(x-1)} e^{^(-t)}dt (Gamma function)
B(x,y) = t^{^(x-1)} (1-t)^{^(y-1)}DT (Beta function) 
Ei(x) = e^{^(-t)}/t DT (exponential integral) or it's variant,NONEQUIVALENT form:

Ei(x) =  + ln(x) + (e^{^t} - 1)/t DT = gamma + ln(x) + (n=1..inf)x^{^n}/(n*n!)

li(x) = 1/ln(t) DT (logarithmic integral)
Si(x) = sin(t)/t DT (sine integral) or it's variant,NONEQUIVALENT form:

Si(x) = sin(t)/t DT = PI/2 - sin(t)/t DT

Ci(x) = cos(t)/t DT (cosine integral) or it's variant,NONEQUIVALENT form:

CI(x) = - COs(t)/t DT = gamma + ln(x) +  (COs(t) - 1) / t DT (cosine integral)

Chi(x) = gamma + ln(x) + (cosh(t)-1)/t DT (hyperbolic cosine integral)
Shi(x) = sinh(t)/t DT (hyperbolic sine integral)
Erf(x) = 2/PI^{^(1/2)}e^{^(-t^2)} DT = 2/PI (n=0..inf) (-1)^{^n}x^{^(2n+1)} / ( n! (2n+1) ) (error function)
FresnelC(x) = COs(PI/2 t^{^2}) DT
FresnelS(x) = sin(PI/2 t^{^2}) DT
dilog(x) = ln(t)/(1-t) DT
Psi(x) = ln(Gamma(x))
Psi(n,x) = n^th derivative of Psi(x)
W(x) = inverse of x*e^{^x}
L _{sub n} (x) = (e^{^x}/n!)( x^{^n} e^{^(-x)} ) ⁽ⁿ⁾ (laguerre polynomial degree n. (n) meaning n^th derivative)
Zeta(s) = (n=1..inf) 1/n^{^s}

Dirichlet's beta function B(x) = (n=0..inf) (-1)^{^n} / (2n+1)^{^x}

                           Derivatives

Table of Derivatives

Power of x.

c = 0

x = 1

xn = n x(n-1)  Proof

Exponential / Logarithmic

ex = ex  Proof

bx = bx ln(b)  Proof

ln(x) = 1/x  Proof

Trigonometric

sin x = cos x  Proof	csc x = -csc x cot x  Proof
cos x = - sin x  Proof	sec x = sec x tan x  Proof
tan x = sec2 x  Proof	cot x = - csc2 x  Proof

Inverse Trigonometric

arcsin x  =  1  (1 - x2)	arccsc x =  -1  |x| (x2 - 1)
arccos x =   -1  (1 - x2)	arcsec x =  1  |x| (x2 - 1)
arctan x =  1  1 + x2	arccot x =  -1  1 + x2

Hyperbolic

sinh x = cosh x  Proof	csch x = - coth x csch x  Proof
cosh x = sinh x  Proof	sech x = - tanh x sech x  Proof
tanh x = 1 - tanh2 x  Proof	coth  x = 1 - coth2 x  Proof

  

Differentiation Identities

Definitions of the Derivative:
df / dx = lim (dx -> 0) (f(x+dx) - f(x)) / dx (right sided)
df / dx = lim (dx -> 0) (f(x) - f(x-dx)) / dx (left sided)
df / dx = lim (dx -> 0) (f(x+dx) - f(x-dx)) / (2dx) (both sided)

 f(t) dt = f(x) (Fundamental Theorem for Derivatives)

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c f(x) = c 
f(x) (c is a constant)

 (f(x) + g(x)) =  f(x) +  g(x) 

 f(g(x)) =  f(g) *  g(x) (chain rule) 

 f(x)g(x) = f'(x)g(x) + f(x)g '(x) (product rule)

 f(x)/g(x) = ( f '(x)g(x) - f(x)g '(x) ) / g^{^2}(x) (quotient rule)

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Partial Differentiation Identities

if f( x(r,s), y(r,s) )

df / dr = df / dx * dx / DR + df / dy * dy / DR

df / ds = df / dx * dx / Ds + df / dy * dy / Ds

if f( x(r,s) )

df / DR = df / dx * dx / DR

df / Ds = df / dx * dx / Ds

directional derivative

df(x,y) / d(Xi sub a) = f1(x,y) cos(a) + f2(x,y) sin(a)

(Xi sub a) = angle counterclockwise from pos. x axis.

 Derivatives: Min, Max, Critical Points...

Asymptotes

Definition of a horizontal asymptote: The line y = y₀ is a "horizontal asymptote" of f(x) if and only if f(x) approaches y₀ as x approaches + or - .Definition of a vertical asymptote: The line x = x₀ is a "vertical asymptote" of f(x) if and only if f(x) approaches + or -  as x approaches x₀from the left or from the right.
Definition of a slant asymptote: the line y = ax + b is a "slant asymptote" of f(x) if and only if lim _(x-->+/-) f(x) = ax + b.

Concavity

Definition of a concave up curve: f(x) is "concave up" at x₀ if and only iff '(x) is increasing at x₀Definition of a concave down curve: f(x) is "concave down" at x₀ if and only if f '(x) is decreasing at x₀
The second derivative test: If f ''(x) exists at x₀ and is positive, then f ''(x)is concave up at x₀. If f ''(x₀) exists and is negative, then f(x) is concave down at x₀. If f ''(x) does not exist or is zero, then the test fails.

Critical Points

Definition of a critical point: a critical point on f(x) occurs at x₀ if and only if either f '(x₀) is zero or the derivative doesn't exist.

Extrema (Maxima and Minima)

Local (Relative) ExtremaDefinition of a local maxima: A function f(x) has a local maximum at x₀ if and only if there exists some interval I containing x₀ such that f(x₀) >= f(x) for all x in I.
Definition of a local minima: A function f(x) has a local minimum at x₀ if and only if there exists some interval I containing x₀ such that f(x₀) <= f(x) for all x in I.
Occurrence of local extrema: All local extrema occur at critical points, but not all critical points occur at local extrema.
The first derivative test for local extrema: If f(x) is increasing (f '(x) > 0) for all x in some interval (a, x₀] and f(x) is decreasing (f '(x) < 0) for all x in some interval [x₀, b), then f(x) has a local maximum at x₀. If f(x) is decreasing (f '(x) < 0) for all x in some interval (a, x₀] and f(x) is increasing (f '(x) > 0) for all x in some interval [x₀, b), then f(x) has a local minimum at x₀.
The second derivative test for local extrema: If f '(x₀) = 0 and f ''(x₀) > 0, then f(x) has a local minimum at x₀. If f '(x₀) = 0 and f ''(x₀) < 0, then f(x) has a local maximum at x₀.
Absolute Extrema
Definition of absolute maxima: y₀ is the "absolute maximum" of f(x) on I if and only if y₀ >= f(x) for all x on I.
Definition of absolute minima: y₀ is the "absolute minimum" of f(x) on I if and only if y₀ <= f(x) for all x on I.
The extreme value theorem: If f(x) is continuous in a closed interval I, then f(x) has at least one absolute maximum and one absolute minimum in I.
Occurrence of absolute maxima: If f(x) is continuous in a closed interval I, then the absolute maximum of f(x) in I is the maximum value of f(x) on all local maxima and endpoints on I.
Occurrence of absolute minima: If f(x) is continuous in a closed interval I, then the absolute minimum of f(x) in I is the minimum value of f(x) on all local minima and endpoints on I.
Alternate method of finding extrema: If f(x) is continuous in a closed interval I, then the absolute extrema of f(x) in I occur at the critical points and/or at the endpoints of I.
(This is a less specific form of the above.)

Increasing/Decreasing Functions

Definition of an increasing function: A function f(x) is "increasing" at a point x₀ if and only if there exists some interval I containing x₀ such that f(x₀) > f(x) for all x in I to the left of x₀ and f(x₀) < f(x) for all x in I to the right of x₀.Definition of a decreasing function: A function f(x) is "decreasing" at a point x₀ if and only if there exists some interval I containing x₀ such that f(x₀) < f(x) for all x in I to the left of x₀ and f(x₀) > f(x) for all x in I to the right of x₀.
The first derivative test: If f '(x₀) exists and is positive, then f '(x) is increasing at x₀. If f '(x) exists and is negative, then f(x) is decreasing at x₀. If f '(x₀) does not exist or is zero, then the test tells fails.

Inflection Points

Definition of an inflection point: An inflection point occurs on f(x) at x₀ if and only if f(x) has a tangent line at x₀ and there exists and interval I containing x₀ such that f(x) is concave up on one side of x₀ and concave down on the other side.

Series (Summation) Expansions

Basic Properties
Convergence Tests

Function-->Summation and Summation-->Function Conversions

Select function f(x) to expand into a summation  f(x) =  ?	Select term an in summation to simplify:  an = ?
Exponential / Logarithm Functions    f(x) = e; e-1; e x    f(x) = ln(x) Root Functions    f(x) = (x); 1/(x)	Geometric Series    an = r n Power Series    an = n; n 2; n 3; ...   an = 1/n; 1/n 2; 1/n 3; 1/n 4; 1/n 5; 1/n 6; 1/n 7; 1/n 8; 1/n 9; 1/n 10; 1/n p