4.7 Antiderivatives
Recall: Differentiation: Given f(x), find f !(x)
Now: Given f(x), find a function whose derivative is f(x). (that is, find F(x) such that F !(x) = f (x) )
F(x) is called an antiderivative of f(x)
EXAMPLE
f(x) = 2x
F(x) =
CHECK
F !(x) =
Antiderivatives are not unique, but they only differ by a constant.
Back to Example
The most general antiderivative of f(x) = 2x is
The most general antiderivative of xn is
EXAMPLES Find the most general antiderivative of each function.
1.
x4
2.
1
x3
3.
3
4.
1
Properties
g(x), then
x
If F(x) is a particular antiderivative of f(x) and G(x) is a particular antiderivative of
1.
2.
a particular antiderivative of cf(x) is cF(x)
a particular antiderivative of f(x) + g(x) is F(x) + G(x)
FUNCTION f(x)
Particular ANTIDERIVATIVE F(x)
xn (n ≠ –1)
1/x
ex
cos x
sin x
sec2 x
sec x tan x
1
1 ! x2
1
1 + x2
EXAMPLE
5 ! 4x 3 + 2x 6
Find the most general antiderivative of g(x) =
x6
EXAMPLES Find f.
1.
f !(x) = 4 / 1 " x 2 , f(1/2) = 1
2.
f !!(x) = 4 " 6x " 40x 3 , f(0) = 2, f !(0) = 1
check: F !(x) = f (x)
EXAMPLE
A particle is moving with the given data:
a(t) = 10 sin t + 3 cos t, s(0) = 0, s(2π) = 12
Find the position of the particle.