4.9 Antiderivatives

antiderivative

F is the antiderivative of f on (a, b) if

F’ = f on (a, b)

most general antiderivative (of f(x)= cos(x))

F(x) = sin x + C

when C is a constant

the reverse power rule

∫ xⁿ dx = 1ⁿ⁺¹ + C

n+1

n =/ -#

addition/ subtraction rule property of indefinite integrals

∫ f(x) ± g(x) dx = ∫ f(x) dx ± ∫ g(x) dx

multiplication/ division rule property of indefinite integrals

∫ C x f(x) dx = C x ∫ f(x) dx

Initial Value Problems

given two pieces of info

• f’(x)

• (a, f(a))

then

• get f(x) +C

• calc C

position, velocity, acceleration

x(t) (position) - d/dt → v(t) (velocity) -d/dt → a(t) (acceleration)