2. Objectives:
At the end of the lesson, the student must be able to
β’ Define antidifferentiation or integration
β’ Be familiar with the properties of indefinite integrals
β’ Perform basic integration by applying the power
formula
β’ Perform integration using simple substitution
3. Antiderivatives
A function πΉ is called an antiderivative (or integral) of the function π on a given interval
if πΉ! π₯ = π π₯ for every value of π₯ in the interval.
For example, the function πΉ π₯ =
"
#
π₯# is an antiderivative of π π₯ = π₯$ on the interval
ββ, +β since for each π₯ in the given interval πΉ! π₯ =
%
%&
"
#
π₯# = π₯$, which is equal
to π π₯ .
However, πΉ π₯ =
"
#
π₯# is not the only antiderivative of π on the given interval. If we add
any constant πΆ to
"
#
π₯# , the function πΊ! π₯ =
%
%&
"
#
π₯# + πΆ = π₯$ = π π₯ .
4. In general, once any single antiderivative is known, the other antiderivatives can
be obtained by adding constants to the known derivatives. Thus,
"
#
π₯#,
"
#
π₯# + 3,
"
#
π₯# + π,
"
#
π₯# + 5 ,
"
#
π₯# β 2 are all antiderivatives of π π₯ = π₯$.
Theorem:
If πΉ π₯ is any antiderivative of π π₯ on an open interval, then for any
constant πΆ, the function πΉ π₯ + πΆ is also an antiderivative on that interval.
Moreover, each antiderivative of π π₯ on the interval can be expressed in the
form πΉ π₯ + πΆ by choosing the constant πΆ appropriately.
5. Definition: The Indefinite Integral
The process of finding antiderivatives is called antidifferentiation or integration.
Thus, if
%
%&
πΉ π₯ = π π₯ , then integrating (or antidifferentiating) the function π π₯
produces an antiderivative of the form πΉ π₯ + πΆ. To emphasize this process, we use
the following notation,
β« π π₯ ππ₯ = πΉ π₯ + πΆ
where,
β« π π₯ ππ₯ is the antiderivative
β« is the integral sign
π π₯ is the integrand
πΆ is the constant of integration
ππ₯ indicates that π₯ is the variable of integration
6. Some of the properties of indefinite integral and basic integration formulas, which need
no proof from the fact that these properties are also known properties of
differentiation, are listed below.
Properties of Indefinite Integral and Basic Integration Formula:
i. β« ππ₯ = π₯ + πΆ ( Definition of an integral)
ii. β« π π π₯ ππ₯ = π β« π π₯ ππ₯ = π πΉ π₯ + πΆ
iii. β« π" π₯ Β± π' π₯ Β± β― Β± π( π₯ ππ₯ = β« π" π₯ ππ₯ Β± β« π' π₯ ππ₯ Β± β― Β± β« π( π₯ ππ₯
iv. β« π₯( ππ₯ =
&!"#
()"
+ πΆ , π β β1 ( The Power Formula )
7. The General Power Formula
In evaluating β« π π π₯ π! π₯ ππ₯ , it will be more convenient to let π’ = π π₯ and write the
differential form ππ’ = πβ² π₯ ππ₯ . Thus,
β« π π’ ππ’ = πΉ π’ + πΆ
The generalized power formula therefore is:
, π π’ "π π π’ =
π π’ "#$
π + 1
+ πΆ π β β1
or
7 π’(ππ’ =
π’()"
π + 1
+ πΆ π β β1
The method of u-substitution may be applied in evaluating an integral with the substitution π’ =
π π₯ and ππ’ = π! π₯ ππ₯