16. Learning Objectives:
1.Illustrate an antiderivative of a function.
2.Compute the general antiderivatives (indefinite
integrals) of polynomial, radical, rational,
exponential, logarithmic, and trigonometric
17. Integral Calculus is the branch of calculus where we study
integrals and their properties. Integration is an essential
concept which is the inverse process of differentiation.
18. Let us have an intuitive approach in finding the
antiderivative of a function. Let us consider this
𝐟′ 𝐱 =
+ 𝟓 =
𝐟′ 𝐱 =
+ 𝟓 = 𝟐 ⋅ 𝒙𝟐−𝟏
𝐟′ 𝐱 =
𝒙𝟐 + 𝟓 = 𝟐𝒙
19. First, we will add one to the exponent of x since
we subtract one from x during the process of
𝐅 𝐱 = 𝟐𝒙𝟏+𝟏
20. Second, we will divide 𝐅 𝐱 = 𝟐𝒙𝟐
by its exponent
2 since we multiply the exponent during the
process of differentiation.
𝐅 𝐱 =
𝑭 𝐱 = 𝒙𝟐
•This operation of determining the original function from
its derivative is the inverse operation of differentiation
and is called antidifferentiation.
•Antidifferentiation is a process or operation that reverses
•Up to this point in Calculus, you have been concerned
primarily with this problem: given a function, find its
•Many important applications of calculus involve the
inverse problem: given the derivative of a function, find
28. Definition of the Antiderivative
A function 𝑭 𝒙 is called an antiderivative
of a function 𝒇 on an interval 𝑰 if 𝑭′ 𝒙 = 𝒇 𝒙
for every value of 𝒙 in 𝑰.
29. Another term for antidifferentiation is
Another term for antiderivative is
30. Integration can be classified into two different
31. The general solution is denoted by
The expression ∫f(x)dx is read as the antiderivative of f with respect to x. So, the differential dx serves
to identify x as the variable of integration. The term indefinite integral is a synonym for
32. Indefinite Integral
Indefinite integrals are not defined using the upper and
lower limits. The indefinite integrals represent the family of
the given function whose derivatives are f, and it returns a
function of the independent variable.
33. From our previous example, the antiderivative of 𝒇′ 𝒙 = 𝟐𝒙
is 𝐅 𝐱 = 𝒙𝟐 + 𝑪, where C is a constant. The derivative of a
constant is zero, so C can be any constant, positive or negative.
The graph of 𝐅 𝐱 = 𝒙𝟐 + 𝑪 is the graph of 𝐅 𝐱 = 𝒙𝟐 shifted
vertically by C units as shown in Figure 1.
34. Definition of the Indefinite Integral
The family of antiderivatives of the function f is called the
indefinite integral of f with respect to x. In symbols, this is
𝒇 𝒙 𝒅𝒙
Thus, if 𝐹 𝑥 is the simplest antiderivative of f and C is any
arbitrary constant, then
𝒇 𝒙 𝒅𝒙 = 𝑭 𝒙 + 𝑪
35. The symbol ∫ is just an elongated S meaning
sum. This integral symbol was devised by
Gottfried Wilhelm Leibniz. The dx refers to the
fact that the function 𝑓 𝑥 is to be
antidifferentiated or integrated with respect to
the variable x.
Note: ∫ 𝒇 𝒙 𝒅𝒙 is read as “the indefinite integral
of 𝑓 𝑥 with respect to x”.
36. Definite Integral
An integral that contains the upper and lower limits
(i.e.) start and end value is known as a definite
integral. The value of x is restricted to lie on a real
line, and a definite Integral is also called a Riemann
Integral when it is bound to lie on the real line.
𝑓 𝑥 𝑑𝑥
38. The Power Rule
If n is any number other than −1, then
𝒏 + 𝟏
In words, when 𝑥𝑛
is integrated, the exponent n of x
is increased by 1 and then 𝑥𝑛+1
is divided by the
new exponent n+1. Notice that the above formula
cannot be used for 𝑛 = −1.