Integration

Integration Formulas

∫ x n dx = xn+1 /(n+1)     if n+1 ≠ 0 ∫1 / x dx = ln |x| ∫ e nx dx = e nx/n     if n ≠ 0

Derivative Formulas

d/dx (xn) = nxn-1 d/dx (ln x) = 1/ x d/dx (e mx) = me mx

Product and Quotient Rules

The Product Rule: d/dx (f(x)g(x)) = f '(x)g(x) + f(x)g '(x) The Quotient Rule: d/dx (f(x)/g(x)) = (f '(x)g(x) - f(x)g '(x))/(g(x)2)

Chain Rules

d/dx (f(u(x))) = d/dx (f(u)) d/dx (u(x)) = f'(u)u'(x)  d/dx (u(x)n) = n u(x)n-1 u'(x) d/dx (ln (u(x)) = u'(x)/ u(x) d/dx (e u(x) ) = e u(x) u'(x)

Change of Variables

du =d/dx (u) dx = u'(x)dx

Integration by Parts

∫u dv = uv - ∫v du

Numerical Integration

∆x = (b-a)/n x0 = a, x1 = x1 + ∆ x , x3 = x2 + ∆x, ... , xn= b. Trapizoidal Approximation for ∫ ab f(x) dx: Tn = 0.5∆x [f(x0) + 2f(x1) + 2f(x2) + ... + 2f(xn-1) + f(xn)] Simpson's Rule (Parabolic Approximation) for ∫ ab f(x) dx: Pn = ∆x [f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + 2f(x4) + ... + 4f(xn-1) + f(xn)]/3

Limit

For n positive : lim x -- > ∞ 1/xn = 0. For n positive : lim x -- > ∞ xn = ∞. For n positive : lim x -- > ∞ 1/enx = 0. For n positive : lim x -- > ∞ enx = ∞. For n positive : lim x -- > - ∞ 1/xn = 0. For n positive : limx -- > - ∞ xn = ±∞.

Maximum and Minimum : 2 Variables

Given a function f(x,y) : The discriminant : D = fxx fyy - fxy2 Decision : For a critical point P= (a,b) If D(a,b) > 0 and fxx(a,b) < 0 then f has a rel-Maximum at P. If D(a,b) > 0 and fxx(a,b) > 0 then f has a rel-Minimum at P. If D(a,b) < 0 then f has a saddle point at P. If D(a,b) = 0 then the test is inconclusive.

Volume and Averager Value (2 variables case.)

Suppose f(x,y) is a function and R is a region on the xy-plane. Assume that f(x,y) is a nonnegative on R. Then the volume under the graph of z = f(x,y) above R is given by Volume = ∫ ∫ R f(x,y) dA Suppose f(x,y) is a function and R is a region on the xy-plane. Then the AVERAGE VALUE of z = f(x,y) over the region R is given by Average Value = ( ∫ ∫ R f(x,y) dA) / (Area of A).

Taylor Polynomial

Given a function f(x) the Taylor Polynomial P n (x) of f(x) around x = a is given by P n (x) = f(a) + f '(a)(x -a) + f ''(a)(x-a)2/2! + f (3)(a)(x-a)3/3! + f (4)(a)(x-a)4/4! + ... + f (n)(a)(x-a)n/n!

Infinite Series

The sum of the Geometric Series a + ar + ar 2 + ... + ar n + ... = a/(1 - r)     if -1 < r < 1     Does Not Converge       Otherwise

Derivative Formulas : Trigonometric Functions

d/dx (sin u) = cos u u'(x) d/dx (cos u) = - sin u u'(x) d/dx (tan u) = sec 2 u u'(x) d/dx (csc u) =- csc (u)cot u u'(x) d/dx (sec u) = sec (u) tan u u'(x) d/dx (cot u) = - csc 2 u u'(x)

Integration Formulas : Trigonometric Functions

∫ sin x dx = -cos x ∫ cos x dx = sin x ∫ tan x dx = - ln |cos x| ∫ sec x dx = ln |sec x + tan x| ∫ csc x dx = ln |csc x -cot x| ∫ cot x = ln |sin x|