5.a – Antiderivatives and The
Indefinite Integral

f  x  dx
Antiderivatives
Definition A function F is called an antiderivative
of f if F ′(x) = f (x) for all x on an interval I.
Theorem If F is an antiderivative of f on an interval
I, then the most general antiderivative of f on I is the
family of functions given by F(x) + c, where c is an
arbitrary constant. F(x) + c is a called a family of
antiderivatives. If c is known, the antiderivative is
called a specific antiderivative.
The Indefinite Integral
Definition The general antiderivative, F(x), of a
function f (x) can be represented by an indefinite
integral
F ( x)   f  x  dx
Like the derivative, the dx denotes the variable with
which we are anti-differentiating.
Examples
Evaluate the indefinite integral (that is, determine the
general anti-derivative) of the flowing functions.
1.  sin x dx
Use number three to
develop a formula for
2.   3e  dt
t
3.   y  5 y  dy
2
3

n
x dx
Properties of the Indefinite Integral
Let c be a constant.
1.

cf  x  dx  ____________________
2.

 f  x   g  x   dx  ____________________
3.

x n dx  _____________________
Note: Always simplify the integrand before evaluating an integral.
Examples
Evaluate the indefinite integral (that is, determine the
general anti-derivative) of the flowing functions.
 2 3x  2

1.   x  1/4  5csc x cot x  dx
x


7 
 t t
2.   3e  5 ln 5 
dt
2 
1 t 

sin  2 y 
3. 
dy
sin y
Examples
Determine f if …
4. f (t )  t  cos t
3
5. f ( x )  2 x  4 ; x  0, f (1)  3
x
4
1

6. f ( x ) 
; f   1
2
1  x2
7. f ( x )  sin x; f (0)  1, f (0)  1, f (0)  1
Example
8. A particle is moving according to the function
a(t) = cos t + sin t [ft/sec2] where s(0) = 0 and
v(0) = 5. Find the position function of this
particle.
Example
9. The graph of a derivative of some function is
given below. Sketch a possible graph of the
function.
(a)
(b)
Table of Basic Indefinite Integrals
Table of Basic Indefinite Integrals
Examples
Evaluate the indefinite integrals.

sec v tan v dv
11.

 3 3/5 
  t  dt
t

12.


10.
13.
e
3 x 2
  6 x  dx Think Carefully 
7 cos  7 y  dy Think Carefully 