Antiderivatives

At this point, we have seen how to calculate derivatives of many functions and have been introduced to a variety of their applications. We now ask a question that turns this process around: Given a function

f,

We answer the first part of this question by defining antiderivatives. The antiderivative of a function

f

Why are we interested in antiderivatives? The need for antiderivatives arises in many situations, and we look at various examples throughout the remainder of the text. Here we examine one specific example that involves rectilinear motion. In our examination in Derivatives of rectilinear motion, we showed that given a position function

s (t)

determining the velocity function requires us to find an antiderivative of the acceleration function. Then, since

v (t) = s^{'} (t),

determining the position function requires us to find an antiderivative of the velocity function. Rectilinear motion is just one case in which the need for antiderivatives arises. We will see many more examples throughout the remainder of the text. For now, let’s look at the terminology and notation for antiderivatives, and determine the antiderivatives for several types of functions. We examine various techniques for finding antiderivatives of more complicated functions later in the text (Introduction to Techniques of Integration).

The Reverse of Differentiation

At this point, we know how to find derivatives of various functions. We now ask the opposite question. Given a function

f,

are also antiderivatives. Are there any others that are not of the form

x^{2} + C

We use this fact and our knowledge of derivatives to find all the antiderivatives for several functions.

Finding Antiderivatives

For each of the following functions, find all antiderivatives.

$f (x) = 3 x^{2}$
$f (x) = \frac{1}{x}$
$f (x) = cos x$
$f (x) = e^{x}$

Because

$\frac{d}{d x} (x^{3}) = 3 x^{2}$

then
$F (x) = x^{3}$
is an antiderivative of
$3 x^{2} .$
Therefore, every antiderivative of
$3 x^{2}$
is of the form
$x^{3} + C$
for some constant
$C,$
and every function of the form
$x^{3} + C$
is an antiderivative of
$3 x^{2} .$
Let $f (x) = ln \| x \| .$
For
$x > 0, f (x) = ln (x)$
and

$\frac{d}{d x} (ln x) = \frac{1}{x} .$

For
$x < 0, f (x) = ln (− x)$
and

$\frac{d}{d x} (ln (− x)) = - \frac{1}{− x} = \frac{1}{x} .$

Therefore,

$\frac{d}{d x} (ln \| x \|) = \frac{1}{x} .$

Thus,
$F (x) = ln \| x \|$
is an antiderivative of
$\frac{1}{x} .$
Therefore, every antiderivative of
$\frac{1}{x}$
is of the form
$ln \| x \| + C$
for some constant
$C$
and every function of the form
$ln \| x \| + C$
is an antiderivative of
$\frac{1}{x} .$
We have

$\frac{d}{d x} (sin x) = cos x,$

so
$F (x) = sin x$
is an antiderivative of
$cos x .$
Therefore, every antiderivative of
$cos x$
is of the form
$sin x + C$
for some constant
$C$
and every function of the form
$sin x + C$
is an antiderivative of
$cos x .$
Since

$\frac{d}{d x} (e^{x}) = e^{x},$

then
$F (x) = e^{x}$
is an antiderivative of
$e^{x} .$
Therefore, every antiderivative of
$e^{x}$
is of the form
$e^{x} + C$
for some constant
$C$
and every function of the form
$e^{x} + C$
is an antiderivative of
$e^{x} .$

Indefinite Integrals

We now look at the formal notation used to represent antiderivatives and examine some of their properties. These properties allow us to find antiderivatives of more complicated functions. Given a function

f,

Given the terminology introduced in this definition, the act of finding the antiderivatives of a function

f

is any real number, is often referred to as the family of antiderivatives of

f .

For some functions, evaluating indefinite integrals follows directly from properties of derivatives. For example, for

n \neq − 1,

Evaluating indefinite integrals for some other functions is also a straightforward calculation. The following table lists the indefinite integrals for several common functions. A more complete list appears in Appendix B.

Integration Formulas
Differentiation Formula	Indefinite Integral
$\frac{d}{d x} (k) = 0$	$\int k d x = \int k x^{0} d x = k x + C$
$\frac{d}{d x} (x^{n}) = n x^{n - 1}$	$\int x^{n} d n = \frac{x^{n + 1}}{n + 1} + C$ for $n \neq − 1$
$\frac{d}{d x} (ln \\| x \\|) = \frac{1}{x}$	$\int \frac{1}{x} d x = ln \\| x \\| + C$
$\frac{d}{d x} (e^{x}) = e^{x}$	$\int e^{x} d x = e^{x} + C$
$\frac{d}{d x} (sin x) = cos x$	$\int cos x d x = sin x + C$
$\frac{d}{d x} (cos x) = − sin x$	$\int sin x d x = − cos x + C$
$\frac{d}{d x} (tan x) = {sec}^{2} x$	$\int {sec}^{2} x d x = tan x + C$
$\frac{d}{d x} (csc x) = − csc x cot x$	$\int csc x cot x d x = − csc x + C$
$\frac{d}{d x} (sec x) = sec x tan x$	$\int sec x tan x d x = sec x + C$
$\frac{d}{d x} (cot x) = − {csc}^{2} x$	$\int {csc}^{2} x d x = − cot x + C$
$\frac{d}{d x} ({sin}^{−1} x) = \frac{1}{\sqrt{1 - x^{2}}}$	$\int \frac{1}{\sqrt{1 - x^{2}}} = {sin}^{−1} x + C$
$\frac{d}{d x} ({tan}^{−1} x) = \frac{1}{1 + x^{2}}$	$\int \frac{1}{1 + x^{2}} d x = {tan}^{−1} x + C$
$\frac{d}{d x} ({sec}^{−1} \\| x \\|) = \frac{1}{x \sqrt{x^{2} - 1}}$	$\int \frac{1}{x \sqrt{x^{2} - 1}} d x = {sec}^{−1} \\| x \\| + C$

it is important to check whether this statement is correct by verifying that

F^{'} (x) = f (x) .

Verifying an Indefinite Integral

Each of the following statements is of the form $\int f (x) d x = F (x) + C .$

Verify that each statement is correct by showing that $F^{'} (x) = f (x) .$

$\int (x + e^{x}) d x = \frac{x^{2}}{2} + e^{x} + C$
$\int x e^{x} d x = x e^{x} - e^{x} + C$

Since

$\frac{d}{d x} (\frac{x^{2}}{2} + e^{x} + C) = x + e^{x},$

the statement

$\int (x + e^{x}) d x = \frac{x^{2}}{2} + e^{x} + C$

is correct.

Note that we are verifying an indefinite integral for a sum. Furthermore,
$\frac{x^{2}}{2}$
and
$e^{x}$
are antiderivatives of
$x$
and
$e^{x},$
respectively, and the sum of the antiderivatives is an antiderivative of the sum. We discuss this fact again later in this section.
Using the product rule, we see that

$\frac{d}{d x} (x e^{x} - e^{x} + C) = e^{x} + x e^{x} - e^{x} = x e^{x} .$

Therefore, the statement

$\int x e^{x} d x = x e^{x} - e^{x} + C$

is correct.

Note that we are verifying an indefinite integral for a product. The antiderivative
$x e^{x} - e^{x}$
is not a product of the antiderivatives. Furthermore, the product of antiderivatives,
$x^{2} e^{x} / 2$
is not an antiderivative of
$x e^{x}$
since

$\frac{d}{d x} (\frac{x^{2} e^{x}}{2}) = x e^{x} + \frac{x^{2} e^{x}}{2} \neq x e^{x} .$

In general, the product of antiderivatives is not an antiderivative of a product.

In [link], we listed the indefinite integrals for many elementary functions. Let’s now turn our attention to evaluating indefinite integrals for more complicated functions. For example, consider finding an antiderivative of a sum

f + g .

—that is, an antiderivative of a sum is given by a sum of antiderivatives. This result was not specific to this example. In general, if

F

From this theorem, we can evaluate any integral involving a sum, difference, or constant multiple of functions with antiderivatives that are known. Evaluating integrals involving products, quotients, or compositions is more complicated (see [link]b. for an example involving an antiderivative of a product.) We look at and address integrals involving these more complicated functions in Introduction to Integration. In the next example, we examine how to use this theorem to calculate the indefinite integrals of several functions.

Evaluating Indefinite Integrals

Evaluate each of the following indefinite integrals:

$\int (5 x^{3} - 7 x^{2} + 3 x + 4) d x$
$\int \frac{x^{2} + 4 \sqrt[3]{x}}{x} d x$
$\int \frac{4}{1 + x^{2}} d x$
$\int tan x cos x d x$

Using [link], we can integrate each of the four terms in the integrand separately. We obtain

$\int (5 x^{3} - 7 x^{2} + 3 x + 4) d x = \int 5 x^{3} d x - \int 7 x^{2} d x + \int 3 x d x + \int 4 d x .$

From the second part of [link], each coefficient can be written in front of the integral sign, which gives

$\int 5 x^{3} d x - \int 7 x^{2} d x + \int 3 x d x + \int 4 d x = 5 \int x^{3} d x - 7 \int x^{2} d x + 3 \int x d x + 4 \int 1 d x .$

Using the power rule for integrals, we conclude that

$\int (5 x^{3} - 7 x^{2} + 3 x + 4) d x = \frac{5}{4} x^{4} - \frac{7}{3} x^{3} + \frac{3}{2} x^{2} + 4 x + C .$
Rewrite the integrand as

$\frac{x^{2} + 4 \sqrt[3]{x}}{x} = \frac{x^{2}}{x} + \frac{4 \sqrt[3]{x}}{x} = 0 .$

Then, to evaluate the integral, integrate each of these terms separately. Using the power rule, we have

$\begin{matrix} \int (x + \frac{4}{x^{2 / 3}}) d x & = \int x d x + 4 \int x^{−2 / 3} d x \\ = \frac{1}{2} x^{2} + 4 \frac{1}{(\frac{−2}{3}) + 1} x^{(−2 / 3) + 1} + C \\ = \frac{1}{2} x^{2} + 12 x^{1 / 3} + C . \end{matrix}$
Using [link], write the integral as

$4 \int \frac{1}{1 + x^{2}} d x .$

Then, use the fact that
${tan}^{−1} (x)$
is an antiderivative of
$\frac{1}{(1 + x^{2})}$
to conclude that

$\int \frac{4}{1 + x^{2}} d x = 4 {tan}^{−1} (x) + C .$
Rewrite the integrand as

$tan x cos x = \frac{sin x}{cos x} cos x = sin x .$

Therefore,

$\int tan x cos x = \int sin x = − cos x + C .$

Initial-Value Problems

We look at techniques for integrating a large variety of functions involving products, quotients, and compositions later in the text. Here we turn to one common use for antiderivatives that arises often in many applications: solving differential equations.

A differential equation is an equation that relates an unknown function and one or more of its derivatives. The equation

is a simple example of a differential equation. Solving this equation means finding a function

y

Sometimes we are interested in determining whether a particular solution curve passes through a certain point

(x_{0}, y_{0})

is an example of an initial-value problem. The condition

y (x_{0}) = y_{0}

is an example of an initial-value problem. Since the solutions of the differential equation are

y = 2 x^{3} + C,

Initial-value problems arise in many applications. Next we consider a problem in which a driver applies the brakes in a car. We are interested in how long it takes for the car to stop. Recall that the velocity function

v (t)

is the derivative of the velocity function. In earlier examples in the text, we could calculate the velocity from the position and then compute the acceleration from the velocity. In the next example we work the other way around. Given an acceleration function, we calculate the velocity function. We then use the velocity function to determine the position function.

Decelerating Car

A car is traveling at the rate of $88$

ft/sec $(60$

mph) when the brakes are applied. The car begins decelerating at a constant rate of $15$

ft/sec².

How many seconds elapse before the car stops?
How far does the car travel during that time?

First we introduce variables for this problem. Let $t$
be the time (in seconds) after the brakes are first applied. Let
$a (t)$
be the acceleration of the car (in feet per seconds squared) at time
$t .$
Let
$v (t)$
be the velocity of the car (in feet per second) at time
$t .$
Let
$s (t)$
be the car’s position (in feet) beyond the point where the brakes are applied at time
$t .$

The car is traveling at a rate of
$88 ft/sec .$
Therefore, the initial velocity is
$v (0) = 88$
ft/sec. Since the car is decelerating, the acceleration is

$a (t) = −15 {ft/s}^{2} .$

The acceleration is the derivative of the velocity,

$v^{'} (t) = −15 .$

Therefore, we have an initial-value problem to solve:

$v^{'} (t) = −15, v (0) = 88 .$

Integrating, we find that

$v (t) = −15 t + C .$

Since
$v (0) = 88, C = 88 .$
Thus, the velocity function is

$v (t) = −15 t + 88 .$

To find how long it takes for the car to stop, we need to find the time
$t$
such that the velocity is zero. Solving
$−15 t + 88 = 0,$
we obtain
$t = \frac{88}{15}$
sec.
To find how far the car travels during this time, we need to find the position of the car after $\frac{88}{15}$
sec. We know the velocity
$v (t)$
is the derivative of the position
$s (t) .$
Consider the initial position to be
$s (0) = 0 .$
Therefore, we need to solve the initial-value problem

$s^{'} (t) = −15 t + 88, s (0) = 0 .$

Integrating, we have

$s (t) = - \frac{15}{2} t^{2} + 88 t + C .$

Since
$s (0) = 0,$
the constant is
$C = 0 .$
Therefore, the position function is

$s (t) = - \frac{15}{2} t^{2} + 88 t .$

After
$t = \frac{88}{15}$
sec, the position is
$s (\frac{88}{15}) \approx 258.133$
ft.

Key Concepts

and then to look for the particular antiderivative that also satisfies the initial condition.

For the following exercises, show that $F (x)$

are antiderivatives of $f (x) .$

F (x) = 5 x^{3} + 2 x^{2} + 3 x + 1, f (x) = 15 x^{2} + 4 x + 3

F^{'} (x) = 15 x^{2} + 4 x + 3

F (x) = x^{2} + 4 x + 1, f (x) = 2 x + 4

F (x) = x^{2} e^{x}, f (x) = e^{x} (x^{2} + 2 x)

F^{'} (x) = 2 x e^{x} + x^{2} e^{x}

F (x) = cos x, f (x) = − sin x

F (x) = e^{x}, f (x) = e^{x}

F^{'} (x) = e^{x}

For the following exercises, find the antiderivative of the function.

f (x) = \frac{1}{x^{2}} + x

f (x) = e^{x} - 3 x^{2} + sin x

F (x) = e^{x} - x^{3} - cos (x) + C

f (x) = e^{x} + 3 x - x^{2}

f (x) = x - 1 + 4 sin (2 x)

F (x) = \frac{x^{2}}{2} - x - 2 cos (2 x) + C

For the following exercises, find the antiderivative $F (x)$

of each function $f (x) .$

f (x) = 5 x^{4} + 4 x^{5}

f (x) = x + 12 x^{2}

F (x) = \frac{1}{2} x^{2} + 4 x^{3} + C

f (x) = \frac{1}{\sqrt{x}}

f (x) = {(\sqrt{x})}^{3}

F (x) = \frac{2}{5} {(\sqrt{x})}^{5} + C

f (x) = x^{1 / 3} + {(2 x)}^{1 / 3}

f (x) = \frac{x^{1 / 3}}{x^{2 / 3}}

F (x) = \frac{3}{2} x^{2 / 3} + C

f (x) = 2 sin (x) + sin (2 x)

f (x) = {sec}^{2} (x) + 1

F (x) = x + tan (x) + C

f (x) = sin x cos x

f (x) = {sin}^{2} (x) cos (x)

F (x) = \frac{1}{3} {sin}^{3} (x) + C

f (x) = 0

f (x) = \frac{1}{2} {csc}^{2} (x) + \frac{1}{x^{2}}

F (x) = - \frac{1}{2} cot (x) - \frac{1}{x} + C

f (x) = csc x cot x + 3 x

f (x) = 4 csc x cot x - sec x tan x

F (x) = − sec x - 4 csc x + C

f (x) = 8 sec x (sec x - 4 tan x)

f (x) = \frac{1}{2} e^{−4 x} + sin x

F (x) = - \frac{1}{8} e^{−4 x} - cos x + C

For the following exercises, evaluate the integral.

\int (−1) d x

\int sin x d x

− cos x + C

\int (4 x + \sqrt{x}) d x

\int \frac{3 x^{2} + 2}{x^{2}} d x

3 x - \frac{2}{x} + C

\int (sec x tan x + 4 x) d x

\int (4 \sqrt{x} + \sqrt[4]{x}) d x

\frac{8}{3} x^{3 / 2} + \frac{4}{5} x^{5 / 4} + C

\int (x^{−1 / 3} - x^{2 / 3}) d x

\int \frac{14 x^{3} + 2 x + 1}{x^{3}} d x

14 x - \frac{2}{x} - \frac{1}{2 x^{2}} + C

\int (e^{x} + e^{− x}) d x

For the following exercises, solve the initial value problem.

f^{'} (x) = x^{−3}, f (1) = 1

f (x) = - \frac{1}{2 x^{2}} + \frac{3}{2}

f^{'} (x) = \sqrt{x} + x^{2}, f (0) = 2

f^{'} (x) = cos x + {sec}^{2} (x), f (\frac{π}{4}) = 2 + \frac{\sqrt{2}}{2}

f (x) = sin x + tan x + 1

f^{'} (x) = x^{3} - 8 x^{2} + 16 x + 1, f (0) = 0

f^{'} (x) = \frac{2}{x^{2}} - \frac{x^{2}}{2}, f (1) = 0

f (x) = - \frac{1}{6} x^{3} - \frac{2}{x} + \frac{13}{6}

For the following exercises, find two possible functions $f$

given the second- or third-order derivatives.

f ″ (x) = x^{2} + 2

f ″ (x) = e^{− x}

Answers may vary; one possible answer is $f (x) = e^{− x}$

f ″ (x) = 1 + x

f ‴ (x) = cos x

Answers may vary; one possible answer is $f (x) = − sin x$

f ‴ (x) = 8 e^{−2 x} - sin x

A car is being driven at a rate of $40$

mph when the brakes are applied. The car decelerates at a constant rate of $10$

ft/sec². How long before the car stops?

5.867

sec

In the preceding problem, calculate how far the car travels in the time it takes to stop.

You are merging onto the freeway, accelerating at a constant rate of $12$

ft/sec². How long does it take you to reach merging speed at $60$

mph?

7.333

sec

Based on the previous problem, how far does the car travel to reach merging speed?

A car company wants to ensure its newest model can stop in $8$

sec when traveling at $75$

mph. If we assume constant deceleration, find the value of deceleration that accomplishes this.

13.75

ft/sec²

A car company wants to ensure its newest model can stop in less than $450$

ft when traveling at $60$

mph. If we assume constant deceleration, find the value of deceleration that accomplishes this.

For the following exercises, find the antiderivative of the function, assuming $F (0) = 0 .$

[T] $f (x) = x^{2} + 2$

F (x) = \frac{1}{3} x^{3} + 2 x

[T] $f (x) = 4 x - \sqrt{x}$

[T] $f (x) = sin x + 2 x$

F (x) = x^{2} - cos x + 1

[T] $f (x) = e^{x}$

[T] $f (x) = \frac{1}{{(x + 1)}^{2}}$

F (x) = - \frac{1}{(x + 1)} + 1

[T] $f (x) = e^{−2 x} + 3 x^{2}$

For the following exercises, determine whether the statement is true or false. Either prove it is true or find a counterexample if it is false.

If $f (x)$

is the antiderivative of $v (x),$

then $2 f (x)$

is the antiderivative of $2 v (x) .$

True

If $f (x)$

is the antiderivative of $v (x),$

then $f (2 x)$

is the antiderivative of $v (2 x) .$

If $f (x)$

is the antiderivative of $v (x),$

then $f (x) + 1$

is the antiderivative of $v (x) + 1 .$

False

If $f (x)$

is the antiderivative of $v (x),$

then ${(f (x))}^{2}$

is the antiderivative of ${(v (x))}^{2} .$

Chapter Review Exercises

True or False? Justify your answer with a proof or a counterexample. Assume that

f (x)

Find the critical points and the local and absolute extrema of the following functions on the given interval.

Determine over which intervals the following functions are increasing, decreasing, concave up, and concave down.

Use Newton’s method to find the first two iterations, given the starting point.

Graph the following functions by hand. Make sure to label the inflection points, critical points, zeros, and asymptotes.

The Reverse of Differentiation

Indefinite Integrals

Initial-Value Problems

Key Concepts

Chapter Review Exercises

Glossary