Calculus 6.1 Antiderivatives and Indefinite Integration

Antiderivatives A function F is an antiderivative of f on an interval I if F’(x) =f(x) for all x in I. A function F is an antiderivative of f on an interval I if F’(x) =f(x) for all x in I. Theorem 6.1: Theorem 6.1: If F is an antiderivative of f on a interval I, then G is an antiderivative of f on the interval I if and only if G is of the form If F is an antiderivative of f on a interval I, then G is an antiderivative of f on the interval I if and only if G is of the form G(x) = F(x) + C where C is a constant. G(x) = F(x) + C where C is a constant.

C is called the constant of integration. C is called the constant of integration. A differential equation in x and y is an equation that involves x, y, and the derivatives of y. A differential equation in x and y is an equation that involves x, y, and the derivatives of y. Example 1: Find the general solution of the differential equation y’=7 Example 1: Find the general solution of the differential equation y’=7

Notations Integral sign  Integral sign  ^ Integrand Integrand The term indefinite integral is a synonym for antiderivative. The term indefinite integral is a synonym for antiderivative.

Basic Integration Rules Integration and differentiation are inverses, so keep in mind that they always undo each other. Integration and differentiation are inverses, so keep in mind that they always undo each other. Example 2: Find the antiderivatives of 6x 2

Example 3

Example 4

Initial Conditions and Particular Solutions If we have information such as this, we can find the determine what the constant of integration is. If we have information such as this, we can find the determine what the constant of integration is. Ex 5: Find the particular solution for f’(s)=6s=8s 3, f(2)=3 Ex 5: Find the particular solution for f’(s)=6s=8s 3, f(2)=3

Vertical Motion a(t)= -9.8m/s 2 a(t)= -9.8m/s 2 a(t)= -32 ft/s 2 a(t)= -32 ft/s 2 Someone cool decides to throw a rock into the Grand Canyon at its deepest point, which is 1800m. The rock is thrown with an initial upward velocity of 21m/s. When would the rock hit the bottom? Someone cool decides to throw a rock into the Grand Canyon at its deepest point, which is 1800m. The rock is thrown with an initial upward velocity of 21m/s. When would the rock hit the bottom?

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