ANTIDERIVATIVES

By:Zerick B. Lucernas

IV-Einstein

Introduction

In each case, the problem is to find a function F whose derivative is a known function f.

If such a function F exists, it is called an antiderivative of f.

Definition

A function F is called an antiderivative of f on an interval I if F’(x) = f (x) for all x in I.

For instance, let f (x) = x2. It is not difficult to discover an antiderivative of f if we keep the Power Rule in mind.

In fact, if F(x) = ⅓ x3, then F’(x) = x2 = f (x).

However, the function G(x) = ⅓ x3 + 100 also satisfies G’(x) = x2. Therefore, both F and G are antiderivatives of f.

Indeed, any function of the form H(x)=⅓ x3 + C, where C is a constant, is an antiderivative of f.

The question arises: Are there any others?

Here we see…

If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is

F(x) + Cwhere C is an arbitrary constant.

Theorem

- f - F

F is an antiderivative of f

Going back to the function f (x) = x2, we see that the general antiderivative of f is ⅓ x3 + C.

Antiderivatives of Power Functions

Fact (The Power Rule)•If f(x) = xr then f′(x) =rxr-1Example:f(x)=x2 then f’(x)?f’(x)= rxr-1

f’(x)= 2x2-1

f’(x)=2x

Rx2-1

By assigning specific values to C, we obtain a family of functions.

Their graphs are vertical translates of one another.

This makes sense, as each curve must have the same slope at any given value of x.

Family of Functions

Family of Functions

• The symbol ∫f(x)dx is traditionally used to represent the most general an antiderivative of f on an open interval and is called the indefinite integral of f .

Notation of Antiderivatives

• Thus, ∫f(x)dx means F’(x) = f (x).

Notation of Antiderivatives

The expression ∫f(x)dx reads:

“the indefinite integral of f with respect to x,” means to find the set of all antiderivatives of f.

Notation of Antiderivatives

( )f x dx

Integral sign

Integrand

x is called the variable of integration

Notation of Antiderivatives

Example: Notation of Antiderivatives

3 32 2because

3 3

x d xx dx C C x

dx

We can write:

Thus, we can regard an indefinite integral as representing an entire family of functions (one antiderivative for each value of the constant C).

Problem

Example -1

32

2

1Evaluate: x + dx

x

32

21

Solution: Let I = x + dxx

6 26 2

1 3= x + +3x + dx

x x

6 26 2

1 3x dx dx 3x dx dx

x x

6 -6 2 -2= x dx+ x dx+3 x dx+3 x dx 7 -5 3 -1x x 3x x

= + + +3× +C7 -5 3 -1

73

5x 1 3

= - +x - +C7 x5x

The final answer

Example -2

1Evaluate: dx

3x+1 - 3x - 11

Solution: Let I = dx3x+1 - 3x - 1

1 3x+1+ 3x - 1= × dx

3x+1 - 3x - 1 3x+1+ 3x - 1

3x+1+ 3x - 1

= dx3x+1 - 3x - 1

3x+1+ 3x - 1= dx

2

1 1= 3x+1 dx+ 3x - 1 dx

2 2

3 32 2

3x+1 3x - 11 1= + +C

3 32 2×3 ×32 2

3 32 2

1 1= 3x+1 + 3x - 1 +C

9 9

Example -2 Cont.

Example -3

xEvaluate: dx

x+2x

Solution: Let I = dxx+2

x+2- 2= dx

x+2x+2 1

= dx - 2 dxx+2 x+2

1 1

-2 2= x+2 dx - 2 x+2 dx

3 12 2

32

x+2 x+2= - 2 +C

12

322

= x+2 - 4 x+2 +C3

Example -4

2Let 5 7 then 10

duu x dx

x

Evaluate

3/ 21

10 3/ 2

uC

3/ 225 7

15

xC

25 7x x dx

2 1/ 215 7

10x x dx u du

Pick u, compute

du

Sub in

Sub in

Integrate

Example -5

Evaluate 4 1 x dxLet 4 1u x

4 du dx

1

4du dx

Solve for dx.

1

21

4

u du3

22 1

3 4u C

3

21

6u C

3

21

4 16

x C

Thank you!!zericklucernas@yahoo.com