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Antiderivative

Function F is antiderivative of F in domain D if F'(x)=f(x) on D

Reimann Sum Definition

Slice interval [a,b] into n equal slices; given function f(x) the n-th RS from x=a to x=b is Sn = f(xi*)Δx + ... +

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Reimann Sum Formula

n

lim ∑ f(x*) Δxn>∞ i=1

Definite Integral

If F is function on interval [a,b], the definite integral of f from a to b is:

a n

∫b f(x)dx = lim ∑ f(xi*)Δx

n>∞ i=1

Provided limits exist and gives same value regardless of choices of the points Xi*; gives signed area under curve

Integral of Constant

b

∫ cdx = c(b-a)

a

Linearity

b

∫ [f(x) +/- g(x)] dx = two separate integrals

a

Comparison

a

f(a) ≥ 0 > ∫ f(x) dx ≥ 0 b

etc.

FTC I

If f is continuous on [a,b] then the unction g is defined by :

x

g(x) = ∫ f(t)dt

a

is continuous and differentiable on (a,b) and g'(x)=fx

FTC II

If f is continuous function and F is antiderivative of f, i.e., F'(x)=f(x), then b

∫ f(x)dx = F(b) - F(a)

a

Indefinite Integral

General antiderivative of f(x)

Substitution Rule

Indefinite integral: if u = g(x) is a differentiable function in D and f is continuous function on D sub in u

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