General Form of an Antiderivative
G(x) = F(x) + CC is the constant of integration
Notation for Antiderivatives
y = f(x)dx = F(x) + C
Initial Conditions and Particlular Solutions
Given: (3x2-1)dx = x3-x+C Passes through (2,4)F(x) = x3-x+C F(2) = 8-2+C F(2) = 4 when C= -2So, particular solution: F(x) = x3-x-2
Sigma Notation
nai = a1 + a2 + a3 +...+ani=1
Important Summation Formulas
Sigma i = [n(n+1)]/2Sigma i2 = [n(n+1)(2n+1)]/6Sigma i3 = [n2(n+1)2]/4
Lower Sum(sum of inscribed rectangles)
s(n) = Sigma f(m1)(change x)mi = 0 + (i-1)(change x)
Upper Sum(sum of circumscribed rectangles)
S(n) = Sigma f(M1)(change x)Mi = 0+i(change x)
Drefinition of the Area of a Region in the Plane
the area of a region bounded by the graph of f, the x-axis, and the vertical lines x=a and x=b is:Area = limSigma f(ci)(change x) where (change x)=[b-a]/n
Definite IntegralFundamental Theorem of Calculus
bSigma f(x)dx = lim Sigma f(ci)(change xi) = F(b)-F(a) = F(x) aa=lower limit; b=upper limit*remember the line thing...
Average Value of a Function
b[1/(b-a)]Sigma f(x)dx = f(C)a
Mean Value Therorem for Integrals
if f is continuous on the closed interval [a,b] then there exixts a number c in the closed interval [a,b] such that:bSigma f(x)dx = f(c)(b-a)a
Second Fundamental Theorem of Calculus
When we defined the definite integral of f on the interval [a,b] we used the constant b as the upper limit of integration and x as the variable of integration. We now look at a slightly different situation in which the variable x is used as the upper limit of integration.(d/dx)Sigma dx
Guidelines for Integration by Substitution
1. Choose a Substitution; choose the inner part of a composite function to sub2. Compute du = g'(x)dx3. Rewrite the integral in terms of the variable u4. Evaluate the resulting integral in terms of u5. Replace u by g(x) to obtain an antiderivative in terms of x6. Check your answer by differentiating
General Power Rule for Integration
Sigma [g(x)]n g'(x)dx = {[[g(x)']n+1]/n+1} +CEquivalently, if u=g(x) then:Sigma undu = [(un+1)/n+1]+C