Section 4.1 – Antiderivatives and Indefinite Integration

Section 4.1 – Antiderivatives and Indefinite Integration

Reversing DifferentiationWe have seen how to use derivatives to solve various

contextual problems. For instance, if the position of a particle is known, then both the velocity and acceleration can be calculated by taking a derivative:

But what if ONLY the acceleration of a particle is known? It would be useful to determine its velocity or its position at a particular time. For this case, a derivative is given and the problem is that of finding the corresponding function.

23 5 7s t t t ' 6 5s t v t t ' 6v t a t Position Function The derivative of the

Position Function is the Velocity Function

The derivative of the Velocity Function is the Acceleration Function

32a t 'v t a t ''s t a tAcceleration

FunctionWhat function has a

derivative of 32?What function has a

second derivative of 32?

Antiderivative

A function F is called an antiderivative of a given function f on an interval I if:

for all x in I.

Example:

'F x f x

2

2

1 1

1

is an antiderivative of

because '

x x

x

F x f x

F x

The Uniqueness of Antiderivatives

Suppose , find an antiderivative of f. That is, find a function F(x) such that .

23f x x 2' 3F x x

3F x xUsing the Power Rule in

Reverse

Is this the only function whose derivative is 3x2?

3

2

5

' 3

H x x

H x x

3

2

11

' 3

K x x

K x x

3

2' 3

M x x

M x x

There are infinite functions whose derivative is 3x2 whose general form is:

3G x x C C is a constant real number (parameter)

Antiderivatives of the Same Function Differ by a Constant

If F is an antiderivative of the continuous function f, then any other antiderivative, G, of f must have the form:

In other words, two antiderivatives of the same function differ by a constant.

G x F x C

Differential EquationA differential equation is any equation that contains derivatives. If a question asks you to “solve a differential equation,” you need to find the original equation (most answers will be in the form y=).

Example:The following is a differential

equation because it contains the derivative of G:

' 2G x x

The general solution to the differential

equation is:

2G x x C

Example 1Find the general antiderivative for the given function.

5f x x

Using the opposite of the Power Rule, a first

guess might be: 6F x x But: 5' 6F x x

Divide this result by 6 to get x5

If: 616F x x Then: 51

6' 6F x x 5x

General Solution: 61

6F x x C


sinf x x

Using the opposite of the Trigonometric Derivatives, a

first guess might be: cosF x x But: ' sinF x x

Multiply this result by -1 to get sinx

If: cosF x x Then: ' sinF x x sin x

General Solution: cosF x x C


35f x x


guess might be: 45F x x But: 3' 20F x x

Divide this result by 4 to get 5x3

If: 454F x x Then: 35

4' 4F x x 35x

General Solution: 45

4F x x C


1x

f x


guess might be: 1 2F x x But: 1 21

2'F x x

Multiply this result by 2 to get x-1/2

If: 1 22F x x Then: 1 212' 2F x x 1 2x

General Solution: 2F x x C

1 21

x

Rewrite if necessary

1 2x


23cos2xf x

Using the opposite trigonometry derivatives:

9 tan 2F x x But: 2' 18sec 2F x x

Multiply this result by 1/2 to get 9sec22x

If: 92 tan 2F x x Then: 29

2' 2 sec 2F x x 29sec 2x

General Solution: 9

2 tan 2F x x C

29

cos 2x

Rewrite if necessary

29sec 2x

Antiderivative NotationThe notation

Means that F is an antiderivative of f. It is called the indefinite integral of f and satisfies the condition that for all x in the domain of f.

f x dx F x C Integral

'F x f x

Variable of Integration

Constant of Integration

Indefinite Integral

New Notation with old ExamplesFind each of the following indefinite integrals.

5.a x dx 616 x C

. sinb x dx cos x C

3. 5c x dx 454 x C

1.x

d dx 2 x C

Basic Integration Rules

Constant Multiple

Sum Rule

Difference Rule

Constant Rule (zero)

cf x dx c f x dx f x g x dx f x dx g x dx

f x g x dx f x dx g x dx

0 dx C

Let f and g be functions and x a variable; a, b, and c be constant; and C is an arbitrary constant.


Constant Rule (non-zero)

Power Rule

Trigonometric Rule

Trigonometric Rule

a dx ax C

cos sinx dx x C


111

n nnx dx x C

sin cosx dx x C


Trigonometric Rule

Trigonometric Rule

Trigonometric Rule

Trigonometric Rule

2sec tanx dx x C

2csc cotx dx x C


sec tan secx x dx x C csc cot cscx x dx x C

Example 1

Evaluate 5 23 7x x dx Sum and Difference

Rules5 23 7x dx x dx dx

Constant Multiple5 23 7x dx x dx dx

Power and Constant Rules

5 1 2 11 15 1 2 13 7x x x C

6 316 7x x x C Simplify

Example 2

Evaluate 5 4sinx x dxRewrite 1 25 4sinx x dx

Sum Rule1 25 4sinx dx x dx

Constant Multiple Rule1 25 4 sinx dx x dx

3 2103 4cosx x C Simplify

Power and Trig Rules 1 2 11

1 2 15 4 cosx x C

Example 3The graph of a certain function F has slope at

each point (x,y) and contains the point (1,2). Find the function F.

34 5x

34 5x dx Difference Rule

34 5x dx dx Constant Multiple Rule34 5x dx dx

4 5x x C Simplify

Power and Constant Rules3 113 14 5x x C

Integrate:

Use the Initial Condition to

find C:

42 1 5 1 C

2 4 C 6 C

4 5 6F x x x

Example 4A particle moves along a coordinate axis in such a way that

its acceleration is modeled by for time t>0. If the particle is at s=5 when t=1 and has velocity v=-2 at this time, where is it when t=4?

32a t t

32v t t dt 32 t dt 3 113 12 t C

22 1 C

2t C Integrate the acceleration to find velocity:

Use the Initial Condition to find C for velocity:

1C

14 4 4 5s

1.25

2 1s t t dt 2 1t dt dt 2 112 1 1t t C

15 1 1 C

1t t C Integrate the Velocity to find position:

Use the Initial Condition to find C for position:

5C

2 1v t t

1 5s t t t Answer the Question:

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Section 4.1 – Antiderivatives and Indefinite Integration