University Mathematics Handbook (2015)
V. Integral Calculus of Single-Variable Functions
Chapter 1. Indefinite Integral
1.1 Antiderivative
a. The function 
is the antiderivative of 
in domain 
, if, for all 
of , there holds 
.
b. If is an antiderivative of , then, for all constant 
, 
is also an antiderivative of .
c. The set of all antiderivatives 
of is called the indefinite integral of . It is denoted
1.2 Properties of Integral
a. 
b. 
c. 
1.3 Immediate Integrals Table
|
a. |
b. |
|
c. |
d. |
|
e. |
f. |
|
g. |
h. |
|
i. |
j. |
|
k. |
l. |
1.4 Integration by Parts
Example: To calculate 
, let's write 
, and 
.
Therefore, 
and 
.
Substituting in the formula, we get
1.5 Substitution Method
a. This way, we substitute variable 
in the function with a function of another variable, 
: 
. This way, the integral in the right side is simpler.
Finding out the integral, we return to variable x. The equality is meaningful if function 
is invertible.
Example: To calculate 
, let's substitute 
. Therefore, 
, and the result is
b. For integrals in the form of 
, the substitution is 
.
c. For integrals in the form of 
, the substitution is 
.
d. For integrals in the form of 
, when 
is a rational function of two-variables, the substitution is 
, from there follows 
. After substitution in the integral, we get a new integral containing no roots.
e. For integrals in the form of 
, the substitution is 
or 
.
f. For integrals in the form of 
, the substitution is 
, 
g. For integrals in the form of 
, the substitution is 
, 
.
1.6 Trigonometric Functions Integration
Calculating the integrals of trigonometric function involves using the relevant trigonometric identities.
a. Integrals
In each of these cases, we should use the relevant trigonometric identities (see II, 2.6 and Chapters 8-10), turning the multiplication into summation.
b. For integral in the form of 
, the substitution is 
.
c. For integral in the form of 
, the substitution is 
.
d. For integral in the form of 
, when 
one or both are even and the other is zero, we use the formulas 
.
e. For integral in the form of 
, when is a rational function, the substitution is 
.
In this case, 
, 
, 
.
Example: To calculate 
, we use the substitution 
.
1.7 Rational Functions Integration
a. Basic Rational Functions of Two Types
1. 
, 
2. 
, 
,
b. Integration of basic rational function of the first type:
1. 
2. 
, 
c. Integration of basic rational function of the second type:
If 
, then, after completing the square to 
and the substitution of 
, 
, we get
d. For a recursive formula for finding 
, , see integrals table in (XVI, 1.4).
e. A General Case
If, in rational function 
, the degree of polynomial 
is higher than or equal to that of polynomial 
, that is, the function is not common, we divide 
by , and the result is the sum of polynomial 
and a common rational function (see X. 3.3)
Step 1: Break down the polynomial into factors of the form of 
and 
, when 
(that is, no real roots exist):
Step 2: Break down the common fraction 
into a sum of unit fractions, the following way:
The computing of quotients 
, 
, 
, is done by adding up all the fractions on the right side into one fraction, the denominator of which is . Then, by equating the denominator of this fraction to the quotients of 
, we get a system of linear equations, with the unknowns , , .
Step 3: Find the integral of the rational function this way:
In the last integral, substitute with the right side of the equality of Step 2. This way, we get the sum of unit fractions integrals.
Example: To find the integral 
, we break down the rational function into unit fractions:
By adding the fractions of the right side, and equating numerators, we get:
After equating quotients, the result is
, 
, 
