ANTIDERIVATIVES

Definition:

reverse operation of finding a derivative

Notice that F is called AN antiderivative and not THE antiderivative.

This is easily understood by looking at the example above.

Some antiderivatives of 𝑓 𝑥 = 4𝑥3 are

𝐹 𝑥 = 𝑥4, 𝐹 𝑥 = 𝑥4 + 2, 𝐹 𝑥 = 𝑥4 − 52

𝑑

𝑑𝑥𝐹(𝑥) = 4𝑥3

Because in each case

Theorem 1:

If a function has more than one antiderivative, then the antiderivatives

differ by a constant.

• The graphs of antiderivatives are vertical translations of each other.

• For example: 𝑓(𝑥) = 2𝑥

Find several functions that are

the antiderivatives for 𝑓(𝑥)

Answer: 𝑥2, 𝑥2 + 1, 𝑥2 + 3, 𝑥2 − 2, 𝑥2 + 𝑐 (𝑐 𝑖𝑠 𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟)

The symbol is called an integral sign,

The function 𝑓 (𝑥) is called the integrand.

The symbol 𝑑𝑥 indicates that anti-differentiation is performed with

respect to the variable 𝑥.

By the previous theorem, if 𝐹(𝑥) is any antiderivative of 𝑓, then

The arbitrary constant C is called the constant of integration.

CxFdxxf )()(

Let f (x) be a function. The family of all functions that are

antiderivatives of f (x) is called the indefinite integral and

has the symbol

dxxf )(

INDEFINITE INTEGRALS

Indefinite Integral

Formulas and Properties

The indefinite integral of a function 𝑓(𝑥) is the family of all functions

that are antiderivatives of 𝑓 (𝑥). It is a function 𝐹(𝑥) whose derivative is 𝑓(𝑥).

Vocabulary:

The definite integral of 𝑓(𝑥) between two limits 𝑎 and 𝑏 is the area

under the curve from 𝑥 = 𝑎 to 𝑥 = 𝑏. It is a number, not a function,

equal to 𝐹(𝑏) − 𝐹(𝑎).

Example 1:

𝑎. 2 𝑑𝑥 = 2𝑥 + 𝐶

𝑏. 16 𝑒𝑡 𝑑𝑡 = 16 𝑒𝑡 + 𝐶

c. 3𝑥4 𝑑𝑥 = 3𝑥5

5+ 𝐶 =

3

5𝑥5 + 𝐶

dxdxxdxxdxxxd 132)132( . 2525

Cxxx

dxdxxdxx

1

33

62132

3625

Cxxx 36

3

1

dxedx

xdxedx

xdxe

xe xxx 4

154

54

5 .

Cex x 4ln5

dxxdxxdxxdxxdx

xxf 43

2

43

2

43

2

32323

2 .

CxxCxx

33

533

5

5

6

33

3

52

Cx

x 3

3

51

5

6

Cxx

dxxxdxx

xxg

2

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8 8

.23

2

2

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Cxx

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Cxx

dxxdxxdxx

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2

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3

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68 .

2

1

3

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2

1

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1

3

Cxx 126 3

4

dxxxxdxxxi 623)3)(2( . 232

Cxxxx

64

234

A differential equation is any equation which contains derivative(s). Solving

a differential equation involves finding the original function from which the

derivative came.

The general solution involves C .

The particular solution uses an initial condition to find the specific value of C.

Definition:

Differential equation is called a separable differential equation

if it is possible to separate 𝑥 and 𝑦 variables.

If

𝑑𝑦

𝑑𝑥 = 𝑓(𝑥)

𝑑𝑦

𝑑𝑥𝑑𝑥 = 𝑓(𝑥) 𝑑𝑥 ⇒ 𝑦 = 𝐹 𝑥 + 𝐶

then the process of finding the antiderivatives of each side of the above

equation (called indefinite integration) will lead to the solution.

Solve the differential equation 𝑑𝑦

𝑑𝑥 = 3𝑥2 if y 2 = −3.

Find both the general and particular solution.

Example:

𝑑𝑦

𝑑𝑥𝑑𝑥 = 3𝑥2 𝑑𝑥

𝑦 = 𝑥3 + 𝐶 general solution:

particular solution: y 2 = −3 ⇒ −3 = 8 + 𝐶 ⇒ 𝐶 = −11

𝑦 = 𝑥3 − 11

INITIAL VALUE PROBLEMS

Particular Solutions are obtained from initial conditions placed on the

solution that will allow us to determine which solution that we are after.

Example:

Find the equation of the curve that passes through (2,6) if its

slope is given by dy/dx = 3x2 at any point x.

The curve that has the derivative of 3x2 is

Since we know that the curve passes through (2, 6), we can find out C

CxCx

dxx

33

2

333

2 26 33 CCCxy

Therefore, the equation is

𝑦 = 𝑥3 − 2