ANTIDERIVATIVES

Definition:

reverse operation of finding a derivative

An antiderivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function .

Notice that F is called AN antiderivative and not THE antiderivative. This is easily understood by looking at the example above.

Some antiderivatives of are

,

𝑑𝑑𝑥

[𝐹 (𝑥) ]=4 𝑥3

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:

Find several functions that are the antiderivatives for

Answer: ,

The symbol is called an integral sign, The function is called the integrand. The symbol indicates that anti-differentiation is performed with respect to the variable .By the previous theorem, if is any antiderivative of 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

1. The indefinite integral of a function is the family of all functions that are antiderivatives of . It is a function whose derivative is

Vocabulary:

2. The definite integral of between two limits and is the area under the curve from to . It is a number, not a function, equal to

We could calculate the function at a few points and add up slices of width Δx like this (but the answer won't be very accurate):

We can make Δx a lot smaller and add up many small slices (answer is getting better):

And as the slices approach zero in width, the answer approaches the true answer.

We now write dx to mean the Δx slices are approaching zero in width.

The area under the curve of a function:

𝐴𝑟𝑒𝑎= lim∆ 𝑥→0

∑𝑖=1

𝑛

𝑓 (𝑥𝑖 ) ∆𝑥=𝑎

𝑏

𝑓 (𝑥 )𝑑𝑥=𝐹 (𝑏)−𝐹 (𝑎)

Example 1:

𝑎 .2𝑑𝑥=2 𝑥+𝐶

𝑏 . 16𝑒𝑡𝑑𝑡=16𝑒𝑡+𝐶c

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

dxxdxxdxxdxxdxx

xf 43

243

2

43

2

32323

2 .

CxxCxx

33

533

5

5

6

33

35

2

Cx

x 3

3

5 1

5

6

Cxx

dxxxdxx

xxg

2

83

8 8

.23

22

34

Cxx

23

43

Cxx

dxxdxxdxx

xh

21

6

34

8 68 6

8 .2

1

3

4

2

1

3

13

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 and variables. If

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 if y. Find both the general and particular solution.

Example:

𝑑𝑦𝑑𝑥

𝑑𝑥=3 𝑥2𝑑𝑥

𝑦=𝑥3+𝐶general solution:

particular solution: y

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

3

32

333

2 26 33 CCCxy

Therefore, the equation is

𝑦=𝑥3−2