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Name: ___________________________________ BC Calculus
5.2 The Indefinite Integral
ANTIDERIVATIVES________________________________________________________________________________________
Given a derivative, can we recover the function that has that derivative? Does such an antiderivative
always exist? Is it unique? What good is it?
Example: Find the derivative of ๐๐
โcos (๐ฅ๐ฅ)
Find the antiderivative of ๐๐(๐ฅ๐ฅ) = sin(๐ฅ๐ฅ). What should we use to describe this โthing?โ
But, we may have more than one antiderivative for function ๐๐(๐ฅ๐ฅ). Why?
Example 1 : ๐๐(๐ฅ๐ฅ) = 2
Possible antiderivatives are: ๐น๐น(๐ฅ๐ฅ) = ___________________
๐น๐น(๐ฅ๐ฅ) = ___________________
___________________
Therefore, the general form for ๐น๐น
is ________________ where ________ is a ___________________.
Antiderivatives
A function ๐น๐น(๐ฅ๐ฅ) is called an antiderivative of a function ๐๐(๐ฅ๐ฅ) on a given open interval if ๐น๐น
โฒ
๐๐(๐ฅ๐ฅ) for all ๐ฅ๐ฅ in the interval.
If ๐น๐น(๐ฅ๐ฅ) is any antiderivative of ๐๐(๐ฅ๐ฅ) on an open interval, then for any constant ๐ถ๐ถ, the function
๐ถ๐ถ is also an antiderivative on that interval. Moreover, each antiderivative of ๐๐(๐ฅ๐ฅ) on the
interval can be expressed in the form ๐น๐น
๐ถ๐ถ by choosing the constant C appropriately.
[๐น๐น(๐ฅ๐ฅ)] = ๐๐(๐ฅ๐ฅ)
THE INDEFINITE INTEGRAL______________________________________________________________________________
The process of finding antiderivatives is called antidifferentiation or integration.
Thus, if (1)
then integrating the function ๐๐(๐ฅ๐ฅ) produces an antiderivative of the form ๐น๐น(๐ฅ๐ฅ) + ๐ถ๐ถ.
To emphasize this process, Equation 1 is recast using integral notation :
where ๐ถ๐ถ is understood to represent an arbitrary constant.
It is important to note that equations 1 and 2 are just different notations to express the same fact.
For example:
The expression โซ ๐๐
๐๐๐ฅ๐ฅ is called an indefinite integral. The adjective โindefiniteโ emphasizes that the
result of antidifferentiation is a โgenericโ function, described only up to a constant term. The โelongated
sโ that appears on the left side of 2 is called an integral sign.
The โpower ruleโ for integration is:
๐๐
SUMMARY___________________________________________________________________________________________________
There are two branches of calculus:
- Integration is the inverse of differentiation.
- Differentiation is the inverse of integration.
PROPERTIES OF INDEFINITE INTEGRALS______________________________________________________________
The first properties of antiderivatives follow directly from the simple constant factor, sum, and difference
rules for derivatives.
One of the most important steps in integration is rewriting the integrand in a form that fits a basic
integration rule.
Example 1:
โซ
dx
x
1
Example 2:
โซ
x + dx
2 2
Example 3:
โซ
dx
x
x
2
3
3
Example 4: x x dx
โซ
(โ 3 sin + 4 cos )
Example 5:
โซ
dx
x
x
2
cos
sin
Example 6:
x
x โ e dx
โซ
Example 7:
2
dx
โ x
โซ
INITIAL CONDITIONS & PARTICULAR SOLUTIONS___________________________________________________
Often enough information is given to determine a particular solution for equation (2). You only need an
initial condition โ a value of ๐น๐น(๐ฅ๐ฅ) for one value of ๐ฅ๐ฅ.
Example 8: Let ๐น๐น(๐ฅ๐ฅ) =
โซ
( 3 x โ 1 ) dx
2
and ๐น๐น
. Find ๐น๐น(๐ฅ๐ฅ).
Example 12: A ball is thrown upward with an initial velocity of 64 feet per second from an initial height
of 80 feet. Use the fact that acceleration due to gravity is โ32 ft/s
2
(a) Find the position function giving the height ๐ ๐ as a function of time ๐ก๐ก.
(b) When does the ball reach its maximum height?
(c) When does the ball hit the ground?
