Math 151
Section 5.7
Antiderivatives
Antiderivative A function F is called the antiderivative of f on an interval I if F !( x ) = f ( x ) for
all x in I.
Example: Is the function F ( x) = x ln x ! x an antiderivative of f ( x) = ln x ?
Example: Find an antiderivative of f ( x) = 2x .
General Antiderivative If F is an antiderivative of f on an interval I, then the most general
antiderivative of f on I is F(x) + C, where C is an arbitrary constant.
Antiderivative Formulas
Function
Particular Antiderivative
Function
c ! f ( x)
f ( x) ± g ( x)
x n , n ! "1
x!1
ekx
cos x
sin x
sec 2 x
sec x tan x
1
x 2 +1
Example: Find the most general antiderivative.
A. f ( x) = 7x 4 + 3x 2 + 7
B. f ( x) = x + 3 x 5 + 34
1
1! x 2
1
x x 2 !1
Particular Antiderivative
Math 151
Example: Find f ( x ) .
A. f !( x) = e4 x + sec x tan x
B. f !( x) =
3
1
4
1
+ 3 + + 3x
4
x
5x
x e
C. f !( x) = "2(1+ x 2 )
"1
Example: Find f ( x ) if f !( x) = 3x 2 +15e3x + 4 and f (0) = 7 .
Example: Find f ( x ) if f !!( x) = 20x 3 + 3sin x , f (0) = 2 , and f !(0) = 8 .
Math 151
Example: A ball is thrown upward with a velocity of 50ft/sec from the edge of a 150-ft building.
A. Find a formula that gives the height of the ball after x seconds.
B. When does the ball reach its maximum height?
C. How fast is the ball traveling when it hits the ground?
Direction Field A direction field (or slope field) is a graphical representation of the slopes of a
function at a variety of points on the xy-plane.
Example: The graph shows the direction field given by f !( x) . Graph f ( x ) for f (0) = 2 and for
f (0) = −4.
Math 151
Antiderivatives of Vector Functions The vector function R (t ) = X (t ),Y (t ) is an antiderivative
of r (t ) = x (t ), y (t ) if R !(t ) = r (t ) .
Example: Find the most general antiderivative of r (t ) = 3t 2 + 2,sec 2 t .
Example: A projectile is fired with an initial speed of 400m/s at an angle of elevation of 60°.
A. Find the position vector of this projectile at time t.
B. Find the range of the projectile.