Full Cheat-Sheet


Derivatives

Definition:
If y = f(x), then
dy
dx
 =
lim f(x + h) - f(x)
h->0 h
  • d/dx [u + v] = du/dx + dv/dx
  • d/dx [u - v] = du/dx - dv/dx
  • d/dx [u*v] = u*v' + v*u'
  • d/dx [u/v] = (v*u' - v*u')/v²
  • d/dx [ f(g(x)) ] = f'(g(x)) * g'(x)
  • For y = axn, dy/dx = naxn-1
  • d/dx [sin x] = cos x
  • d/dx [cos x] = sin x
  • d/dx [tan x] = sec²x
  • d/dx [csc x] = - (csc x)(cot x)
  • d/dx [sec x] = (sec x)(tan x)
  • d/dx [cot x] = - csc²x
Integration

Fundamental Theorem of Calculus:
Where d/dx [ F(x) ] = f(x),
ò ab f(x) dx = F(b) - F(a)
(The integral from a to b of a function is equal to the function's antiderivative at b minus that at a.)
 
Some Common Indefinite Integrals
  • ò (axn)dx = (axn+1)/(n+1) + C
  • ò 1/x dx = ln|x| + C
  • ò ex dx = ex + C
  • ò bx dx = bx / ln(b) + C
  • ò ln(x) dx = x ln(x) - x + C
  • ò sin x dx = -cos x + C
  • ò cos x dx = sin x + C
  • ò tan x dx = -ln|cos x| + C
  • ò csc x dx = - ln|csc x + cot x| + C
  • ò sec x dx = ln|sec x + tan x| + C
  • ò cot x dx = ln|sin x| + C
  • ò sec² x dx = tan x + C
  • ò csc x cot x dx = - csc x + C
  • ò sec x tan x dx = sec x + C
  • ò csc² x dx = - cot x + C
Methods of Integration
If you cannot simply integrate using known combinations, the following methods also exist:
  • Integration by substitution: a method by which some variable is substituted for part of the function f(x) (generally some complicated function within the larger function) in order to make integration simpler. Take, for example, the integral
    ò 3 cos(5x) dx
    This would ordinarily not be easy to integrate. Now, set
    u = 5x
    From here, determine the value of dx and substitute the new equations back into the original.
    u = 5x
    du = 5 dx
    dx = du/5

    ò 3 cos(u) (du/5)
    ò (3/5)cos(u) du = (3/5)sin(u) + C
    (3/5)sin(u) + C = (3/5)sin(5x) + C


  • Integration by parts: a method of integrating two functions multiplied together (the opposite of the product rule for derivatives), following the formula:
    ò u(dv/dx) dx = uv - òv(du/dx) dx

    An example is as follows:
    ò x sin(x) dx = ?
    u = xdv/dx = sin(x)
    du/dx = 1v = -cos(x)
    ò u(dv/dx) dx= uv - òv(du/dx) dx
    ò x sin(x) dx= -x cos(x) - ò -cos(x)(1) dx
      = -x cos(x) - (-sin(x))
    ò x sin(x) dx= -x cos(x) + sin(x)


  • Integration by partial fractions: For rational functions in which the denominator has a higher degree than the numerator, integration can be made easier by splitting the integrand into two partial fractions. (Partial fractions are fractions which can be added to result in another fraction: for example, the partial fractions of 5/6 are 1/2 and 1/3, because 1/2 + 1/3 = 5/6.)
    An example of this method follows:
    ò
    ___x + 3___
    x² + 3x + 2
    		 dx   =   ò
    __2__
    x + 1
    		 dx   -   ò
    __1__
    x + 2
    		 dx


  • Numerical Integration: When symbolic methods fail, use of some numerical approximation method will give useful answers along a specified interval. Most calculators enact these methods to give extremely exact answers by using very tiny subdivisions.

    • RAM: Rectangular Approximation Method.
      ò @ Si yi(dx) where yi represents the heights of the successive rectangles and dx is the common width of the rectangles (the division).
    • TRAPPROX: Trapezoidal approximation.
      ò @ ½(dx)[y0 + 2(y1 + ... + yn-1) + yn]
    • Simpson's rule: Fitting parabolic segments beneath a curve.
      ò @ (b-a)[y0 + 4y1 + 2y2 + 4y3 + 2y4 + ... + 2yn-2 + 4yn-1 + yn]/3n
      where the interval is between a and b and n is an (even) number of subdivisions. (With higher number of subdivisions, smaller individual divisions are achieved.)