Definition: A function F(x) is called an antiderivative of f(x) on an interval I, if F'(x) = f(x) for all x in I.

Theorem: If F is an antiderivative of f on I, then the most general antiderivative of f is

F(x)+c (This says that antiderivatives are unique, up to a constant.)

Fact: Every differentiation rule leads to an anti-derivative result, for example

d/dx x ⁿ = n x ^n-1 imples that

x ⁿ + C is an anti-derivative of n x ^n-1

or d/dx tan ^-1(x) = 1 / (1+x²) implies that

the antiderivative of 1 / (1+x²) is tan ^-1(x) + C

Note : One can find anti-derivatives term by term (since differentiation is linear!)

Note: : One can repeatedly anti-differentiate - If F''(x) = f(x) then the anti-derivative of the anti-derivative of f is given by F(x) + C₁ x + C₂

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Antiderivatives and Direction Fields:

A direction field is a vector field whose direction at the point < x,y > is given by m=f'(x). An example is given below:

Connecting the "arrows" gives you a curve which is the solution of the differential equation y'=f(x), that is a set of antiderivatives.

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Vector valued functions have anti-derivatives also:

Definition A vector valued function R (t) = X(t) i + Y(t) j is called an antiderivative of r(t) = x(t) i + y(t) j if R'(t) = r'(t) for all t in I, that is X'(t) = x(t) and Y'(t)=y(t).

Corollary: R(t) + C is the most general antiderivative, where C = C₁i + C₂j.

Last modified Mon Nov 18 15:06:07 CST 1996