Antiderivatives

1. Lesson 1
Antiderivatives and the
Power Formula

2. Objectives:
At the end of the lesson, the student must be able to
• Define antidifferentiation or integration
• Be familiar with the properties of indefinite integrals
• Perform basic integration by applying the power
formula
• Perform integration using simple substitution

3. Antiderivatives
A function 𝐹 is called an antiderivative (or integral) of the function 𝑓 on a given interval
if 𝐹! 𝑥 = 𝑓 𝑥 for every value of 𝑥 in the interval.
For example, the function 𝐹 𝑥 =
"
#
𝑥# is an antiderivative of 𝑓 𝑥 = 𝑥$ on the interval
−∞, +∞ since for each 𝑥 in the given interval 𝐹! 𝑥 =
%
%&
"
#
𝑥# = 𝑥$, which is equal
to 𝑓 𝑥 .
However, 𝐹 𝑥 =
"
#
𝑥# is not the only antiderivative of 𝑓 on the given interval. If we add
any constant 𝐶 to
"
#
𝑥# , the function 𝐺! 𝑥 =
%
%&
"
#
𝑥# + 𝐶 = 𝑥$ = 𝑓 𝑥 .

4. In general, once any single antiderivative is known, the other antiderivatives can
be obtained by adding constants to the known derivatives. Thus,
"
#
𝑥#,
"
#
𝑥# + 3,
"
#
𝑥# + 𝜋,
"
#
𝑥# + 5 ,
"
#
𝑥# − 2 are all antiderivatives of 𝑓 𝑥 = 𝑥$.
Theorem:
If 𝐹 𝑥 is any antiderivative of 𝑓 𝑥 on an open interval, then for any
constant 𝐶, the function 𝐹 𝑥 + 𝐶 is also an antiderivative on that interval.
Moreover, each antiderivative of 𝑓 𝑥 on the interval can be expressed in the
form 𝐹 𝑥 + 𝐶 by choosing the constant 𝐶 appropriately.

5. Definition: The Indefinite Integral
The process of finding antiderivatives is called antidifferentiation or integration.
Thus, if
%
%&
𝐹 𝑥 = 𝑓 𝑥 , then integrating (or antidifferentiating) the function 𝑓 𝑥
produces an antiderivative of the form 𝐹 𝑥 + 𝐶. To emphasize this process, we use
the following notation,
∫ 𝑓 𝑥 𝑑𝑥 = 𝐹 𝑥 + 𝐶
where,
∫ 𝑓 𝑥 𝑑𝑥 is the antiderivative
∫ is the integral sign
𝑓 𝑥 is the integrand
𝐶 is the constant of integration
𝑑𝑥 indicates that 𝑥 is the variable of integration

6. Some of the properties of indefinite integral and basic integration formulas, which need
no proof from the fact that these properties are also known properties of
differentiation, are listed below.
Properties of Indefinite Integral and Basic Integration Formula:
i. ∫ 𝑑𝑥 = 𝑥 + 𝐶 ( Definition of an integral)
ii. ∫ 𝑐 𝑓 𝑥 𝑑𝑥 = 𝑐 ∫ 𝑓 𝑥 𝑑𝑥 = 𝑐 𝐹 𝑥 + 𝐶
iii. ∫ 𝑓" 𝑥 ± 𝑓' 𝑥 ± ⋯ ± 𝑓( 𝑥 𝑑𝑥 = ∫ 𝑓" 𝑥 𝑑𝑥 ± ∫ 𝑓' 𝑥 𝑑𝑥 ± ⋯ ± ∫ 𝑓( 𝑥 𝑑𝑥
iv. ∫ 𝑥( 𝑑𝑥 =
&!"#
()"
+ 𝐶 , 𝑛 ≠ −1 ( The Power Formula )

7. The General Power Formula
In evaluating ∫ 𝑓 𝑔 𝑥 𝑔! 𝑥 𝑑𝑥 , it will be more convenient to let 𝑢 = 𝑔 𝑥 and write the
differential form 𝑑𝑢 = 𝑔′ 𝑥 𝑑𝑥 . Thus,
∫ 𝑓 𝑢 𝑑𝑢 = 𝐹 𝑢 + 𝐶
The generalized power formula therefore is:
, 𝑓 𝑢 "𝑑 𝑓 𝑢 =
𝑓 𝑢 "#$
𝑛 + 1
+ 𝐶 𝑛 ≠ −1
or
7 𝑢(𝑑𝑢 =
𝑢()"
𝑛 + 1
+ 𝐶 𝑛 ≠ −1
The method of u-substitution may be applied in evaluating an integral with the substitution 𝑢 =
𝑔 𝑥 and 𝑑𝑢 = 𝑔! 𝑥 𝑑𝑥

8. Examples:
1. Evaluate ∫ 6𝑥' − 4𝑥 + 5 𝑑𝑥
Solution:
∫ 6𝑥' − 4𝑥 + 5 𝑑𝑥 = 6 ∫ 𝑥' 𝑑𝑥 − 4 ∫ 𝑥 𝑑𝑥 + 5 ∫ 𝑑𝑥
= 6 ;
&$
$
− 4 ;
&%
'
+ 5𝑥 + 𝐶
= 2𝑥$ − 2𝑥' + 5𝑥 + 𝐶
2. Evaluate ∫ 4 𝑡 +
'
*
𝑑𝑡
Solution:
∫ 4 𝑡 +
'
*
𝑑𝑡 = 4 ∫ 𝑡
#
% 𝑑𝑡 + 2 ∫
"
*
#
%
𝑑𝑡 = 4 ∫ 𝑡
#
% 𝑑𝑡 + 2 ∫ 𝑡+
#
% 𝑑𝑡
= 4 ;
*
$
%
$
%
+ 2 ;
*
#
%
#
%
+ 𝐶 =
,
$
𝑡
$
% + 4𝑡
#
% + 𝐶

9. 3. Evaluate ∫ 3𝑥 + 1 ' 𝑑𝑥
Solution 1:
∫ 3𝑥 + 1 ' 𝑑𝑥 = ∫ 9𝑥' + 6𝑥 + 1 𝑑𝑥
= 9 ∫ 𝑥' 𝑑𝑥 + 6 ∫ 𝑥 𝑑𝑥 + ∫ 𝑑𝑥
= 9 ;
&$
$
+ 6 ;
&%
'
+ 𝑥 + 𝐶
= 3 𝑥$ + 3𝑥' + 𝑥 + 𝐶
Solution 2:
∫ 3𝑥 + 1 ' 𝑑𝑥 =
"
$
∫ 3𝑥 + 1 ' ; 3 𝑑𝑥
=
"
$
;
$&)" $
$
+ 𝐶
=
"
-
3𝑥 + 1 $ + 𝐶

10. Solution 3: (by u-substitution)
∫ 3𝑥 + 1 ' 𝑑𝑥 =
"
$
∫ 𝑢' 𝑑𝑢
=
"
$
;
.$
$
+ 𝐶
=
"
-
3𝑥 + 1 $ + 𝐶
4. Evaluate ∫
&$ %&
'&&+/ '
Solution:
∫
&$ %&
'&&+/ ' = ∫ 2𝑥# − 5 +0 𝑥$𝑑𝑥
=
"
,
∫ 𝑢+0 𝑑𝑢
=
"
,
;
.()
+1
+ 𝐶 = −
"
#, '&&+/ ) + 𝐶

11. 5. Evaluate ∫
')$ 234 5 & %5
678%5
Solution:
∫
')$ 234 5 & %5
678%5
= ∫ 2 + 3 tan 𝑦 # 𝑠𝑒𝑐'𝑦 𝑑𝑦
=
"
$
∫ 𝑢# 𝑑𝑢
=
"
$
;
.*
/
+ 𝐶 =
"
"/
2 + 3 tan 𝑦 / + 𝐶
6. Evaluate ∫ cot 𝑥 ln sin 𝑥 𝑑𝑥
Solution:
∫ cot 𝑥 ln sin 𝑥 𝑑𝑥 = ∫ 𝑢 𝑑𝑢
=
.%
'
+ 𝐶
=
9(% :;4 &
'
+ 𝐶

12. 7. Evaluate ∫
#&+" %&
'&)$ * &+' *
Solution:
∫
#&+" %&
'&)$ * &+' * = ∫
#&+" %&
'&)$ &+' *
= ∫
#&+" %&
'&%+&+1 *
= ∫ 2𝑥' − 𝑥 − 6 +/ 4𝑥 − 1 𝑑𝑥
= ∫ 𝑢+/ 𝑑𝑢
=
.(&
+#
+ 𝐶
=
'&%+&+1
(&
+#
+ 𝐶
= −
"
# '&%+&+1 & + 𝐶

13. Exercises:
Evaluate the following integrals:
1. ∫ 𝑡 −
$
𝑡' +
*
𝑡$ 𝑑𝑡 2. ∫
234 '& %&
$)<4 =>: '& )

14. Exercises:
3. ∫
")$?(%+ ,
%&
?%+ 4. ∫ cot 3𝑥 𝑐𝑠𝑐$3𝑥 𝑑𝑥

15. Exercises:
5. 7
2𝑥 + 1 𝑑𝑥
𝑥 + 4 1 𝑥 − 3 1 6. 7 𝑠𝑖𝑛# 𝑒$& cos 𝑒$& 𝑒$& 𝑑𝑥