In this section we learn to compute general anti-derivatives, also known as indefinite integrals.
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.
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 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 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.
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.
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.
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.
because the derivative of is , by the chain rule, and hence the derivative of is , by
the constant multiple rule.
(problem 10)
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.
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).
example 13 Compute
We use the Power Rule with () and we get The answer can be confirmed by
observing that the derivative of is
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).
Special Case
The Power Rule does not work when so we consider this as a special case: