Antiderivatives

Indefinite Integral

The family of antiderivatives of a function f indicated by

The symbol is a stylized S to indicate summation

2

( )f x dx

Indefinite Integral

The indefinite integral is a family of functions

The + C represents an arbitrary constant• The constant of integration

3

3 41

4x dx x C

2 13 4 3 4x dx x x C

Properties of Indefinite Integrals

The power rule

The integral of a sum (difference) is the sum (difference) of the integrals

4

( ) ( ) ( ) ( )f x g x dx f x dx g x dx

11, 1

1n nx dx x C n

n

Properties of Indefinite Integrals

The derivative of the indefinite integral is the original function

A constant can be factored out of the integral

5

( ) ( )d

f x dx f xdx

( ) ( )f x dx xk k f x d

• Example : Evaluate (5x 4x3 )dx.

(5x 4x3)dx 5x dx 4x3dx 5 x dx 4 x3dx 5

x2

2 4

x4

4C

5

2x2 x4 C

Examples

Determine the indefinite integrals as specified below

7

5x dx 4 2x x dx 7 dx12x dx

Integrate

dttt 13

dxx33 dxxx 872 2

Find each antiderivative

dx

xxx

23

5

4

3 25

dxx

1 dxx21

dx

xx 43

54 dxx

Find each antiderivative

dyyy 43 42

dxxx

321 dxx dxx 232

dx

x

xx4

2 13

Find each antiderivative

dyx

xx )23)(52( dxxsin4

dxx

3

cos2 dxx2cos

5 dxxx )sin9cos4(

Find each antiderivative

dxx

x2cos

sin d22 csc2

Solve the differential equation

3)2(,23)(' fxxf

Solve the differential equation

1)3(,22)(' 2 fxxxf

Solve the differential equation

2)1(,1)4(',2)('' ffxf

Solve the differential equation

1)2(,30)5(',2)('' ffxxf

Given that the graph of f(x) passes through the point (1,6)

and that the slope of its tangent line at (x.f(x) is 2x+1, find f(6)