5.1 Integrals Rita Korsunsky

Definition A function F is an antiderivative of f on an interval I if for every x in I ……………

Indefinite integral because

y When C = 0 C = -1 2 C = 1 C = -2 1 x C = 2 -1 -2

Examples

Theorem Proof:

Properties of the Integral for any constant c Proof:

Example Evaluate

Evaluate Example

Example Evaluate

Example Find the equation of the curve that passes through (1,-1) and whose slope at any point (x,y) is 3x 2 Step One – Integrate Step Two - Substitute initial conditions Step Three - Solve for C Step Four - Substitute C back into general solution equation

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