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Mathematical Expression Editor
In this section we learn to compute general anti-derivatives, also known as indefinite
integrals.
1 Anti-derivatives
In a typical differentiation problem, we are given a function and we are asked to
compute the derivative, . In a typical anti-differentiation problem, we are given the
derivative of a function, and we are asked to compute the original function .
Anti-differentiation is thus the reverse process of differentiation. There is a small
twist, however as we will see in our first example.
example 1 Given that , find . By now, we are all familiar with the differentiation formula: and this leads us the the
answer: So, the anti-derivative of (with respect to ) is . But wait- notice that the
derivative of is also . so is also a solution to this anti-differentiation problem. We
now recognize that there are actually infinitely many solutions to this problem,
because if is any constant, then Thus the answer to the anti-differentiation problem
is actually not just a single function but a family of functions each pair of which
differing by a constant. The graph below shows several members of this family.
Anti-differentiation problems arise naturally in projectile motion scenarios.
example 2 Suppose the velocity of a projectile is given by the equation . Find the
position function . We recall from previous discussions of rectilinear motion, of which projectile motion
is a special case, that Hence to recover from as in this problem, we must
anti-differentiate . Restating the problem in terms of differentiation, we need to find
the function such that . Our familiarity with differentiating polynomials leads us to
guess We can verify that this is a correct answer by differentiating it: But let us not
forget the lesson of the previous example: there is more than one answer. The answer
is a family of functions, each pair of which differing by a constant. So a complete
solution set to our problem is Note that the constant in this equation represents the
position of the object at time . We denote this particular constant as (rather
than ), the initial position of the object and we can write our answer as
.
We see from the previous example that in order to know the exact position of
the object, we need more information than just the velocity function. If
we were also given the initial position of the object (or the position at any
time), then we can find the precise position function. We do this in the next
example.
example 3 Suppose the velocity of a projectile (in ft/sec) is given by the equation .
Find the height of the projectile at time seconds, given that the projectile lands at
time seconds.
We recall from previous example that The problem requires us to find and in order
to do this, we must first find , the initial launch height of the projectile. We are given
that the object lands after three seconds, so . We can use this to find by plugging
into the position function: Solving the second equation for yields ft. Thus
the position function is Finally, we can find the position at time seconds
(problem 3a) Given , find the family of anti-derivatives. Don’t forget to add to your
answers.
(problem 3b) Find the precise anti-derivatives satisfying the given initial condition:
(problem 3c) Find the height of the projectile at time seconds if the total time in
the air is 5 seconds and the velocity at time seconds is given in ft/sec as
. The initial height of the projectile is ft. The position at time is given by . The height at time seconds is ft.
It turns out that the process of anti-differentiation is directly tied to finding the area
under a curve. We will study the area problem in detail in an upcoming lesson, but
for now, we will learn an important notation that is used in the discussion of the area
problem.
2 The Indefinite Integral
The indefinite integral is denoted by and it is read “the integral of dx.” To
compute an indefinite integral we need to find all anti-derivatives, of . Notice the
change in the notation from and to and . The process of computing an
integral is called integration in addition to anti-differentiation. Since any two
(continuous) functions that have the same derivative differ by a constant,
if we can find one example of a function whose derivative is , i.e., , then
we can write where C is any constant. The symbol is called the integral
sign, the function is called the integrand and is called the constant of
integration.
3 Examples of Basic Indefinite Integrals
A basic indefinite integral is one that can be computed either by recognizing the
integrand as the derivative of a familiar function or by reversing the Power Rule for
Derivatives.
example 4 because the derivative of is .
(problem 4a)
Do not add the +C to your answer
(problem 4b)
The answer is a minor modification of
Do not add the +C to your answer
example 5 because the derivative of is .
(problem 5a) Compute
Do not add the +C to your answer
(problem 5b) Compute
The answer is a minor modification of
Do not add the +C to your answer
example 6 because the derivative of is .
(problem 6) Compute
Do not add the +C to your answer
example 7 because the derivative of is .
because the derivative of is , by the chain rule, and hence the derivative of is , by
the constant multiple rule.
(problem 7a) Compute
Do not add the +C to your answer
(problem 7b) Compute
The answer is a minor modification of
Do not add the +C to your answer
(problem 7c) Compute
The answer is a minor modification of
Do not add the +C to your answer
example 8 because the derivative of is .
(problem 8) Compute
Do not add the +C to your answer
example 9 because the derivative of is .
(problem 9) Compute
Note that the variable is
Write your answer as to a power.
Do not add the +C to your answer
example 10 because the derivative of is .
(problem 10) Compute
Note that the variable is
Do not add the +C to your answer
(problem 10)
example 11 because the derivative of is .
(problem 11) Compute
Note that the variable is
Do not add the +C to your answer
4 Power Rule for Indefinite Integrals
The Power Rule for Indefinite Integrals reverses the Power Rule for Derivatives.
Instead of subtracting 1 from the exponent, we add 1 and instead of multiplying by
the exponent, we divide.
Power Rule for Indefinite Integrals where is any number except .
example 12 Compute
We use the Power Rule with and we get The answer can be confirmed by observing
that the derivative of is (using the Constant Multiple Rule and the Power Rule for
Derivatives).
(problem 12a) Compute
Use the Power Rule with
The Power Rule says C
Do not add the +C to your answer
(problem 12b) Compute
Use the Power Rule with
The Power Rule says C
Do not add the +C to your answer
example 13 Compute
We use the Power Rule with () and we get The answer can be confirmed by
observing that the derivative of is
(problem 13a) Compute
Use the Power Rule with
The Power Rule says C
Note that the variable is
Do not add the +C to your answer
(problem 13b) Compute
Use the Power Rule with
The Power Rule says C
Note that the variable is
Do not add the +C to your answer
example 14 Compute
We use the Power Rule with and we get The answer can be confirmed by observing
that the derivative of is (using the constant multiple rule and the Power Rule for
Derivatives).
(problem 14a) Compute
Use the Power Rule with
The Power Rule says C
Note that the variable is
Do not add the +C to your answer
(problem 14b) Compute
Use the Power Rule with
The Power Rule says C
Note that the variable is
Do not add the +C to your answer
example 15 Compute
We rewrite as and we use the Power Rule with and we get \begin{align*} \int \sqrt x \ dx &= \int x^{1/2} \ dx\\ &= \frac{x^{3/2}}{3/2} + C\\ &= \tfrac{2}{3}x^{3/2} + C\\ &= \tfrac 23 \sqrt{x^3} + C\\ &= \tfrac 23 x\sqrt{x} + C. \end{align*}
(problem 15a) Compute
Rational exponents:
Use the Power Rule with
The Power Rule says C
Do not add the +C to your answer
(problem 15b) Compute
Rational exponents:
Use the Power Rule with
The Power Rule says C
Do not add the +C to your answer
example 16 Compute
We rewrite as and use the Power Rule with . We get \begin{align*} \int \frac{1}{x^3} \ dx &= \int x^{-3} \ dx \\ &= \frac{x^{-2}}{-2} + C\\ &= -\tfrac{1}{2}x^{-2} + C \\ &= -\tfrac{1}{2}\cdot \frac{1}{x^2} + C \\ &= -\frac{1}{2x^2} +C. \end{align*}
(problem 16a) Compute
Negative exponents:
Use the Power Rule with
The Power Rule says C
Do not add the +C to your answer
(problem 16b) Compute
Negative exponents:
Use the Power Rule with
The Power Rule says C
Do not add the +C to your answer
example 17 Compute
We rewrite as and use the Power Rule with . We get \begin{align*} \int \frac{1}{\sqrt x} \ dx &= \int x^{-1/2} \ dx \\ &= \frac{x^{1/2}}{1/2} + C \\ &= 2x^{1/2} + C \\ &= 2\sqrt x +C. \end{align*}
(problem 17a) Compute
Negative rational exponents:
Use the Power Rule with
The Power Rule says C
Do not add the +C to your answer
(problem 17b) Compute
Negative rational exponents:
Use the Power Rule with
The Power Rule says C
Do not add the +C to your answer
Special Case
The Power Rule does not work when so we consider this as a special case:
example 18 We can verify this by noting that the derivative of is .