4.5 Integration By Pattern Recognition

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Integration by Pattern Recognition:

Cn

uduu

nn

1

1

The first basic type of integration problem is in the form:

Integrate by recognizing the Pattern duu n

dxxdu 26 Then

Therefore, this integral is of the type: duu 4

dxxx 243 6)52(

Note: If )52( 3 xuIntegrating we get:

Cu

duu 5

54

Substitute, )52( 3 xu

Henceforth, Cx

dxxx

5

)52(6)52(

53243

)52( 3 xuBut,

dxx 2)32( 21

Note: If )32( xudxdu 2Then

Note: This is exactly in the form! duun

dxx 232

Therefore, this integral is of the type: duu 2

1Integrating we get:

23

23

21 uu

)32( xuBut, Substitute, )32( xu

Henceforth, Cxdxx 23

21

323

22)32(

Multiplying by a Form of 1 to integrate:

xdxdu 2Then

Note: This is not exactly in the form! duunThe inside of the Integral has to be multiplied by 2

Therefore the outside of the Integral has to be multiplied by ½, since( 2) (½) = 1, and as long as we multiple the entire integral by a numeric form of 1 we can proceed with integration.

xdxx 22 )5(

Note: If )5( 2 xu

Now multiply by a form of 1 to integrate:

xdxdu 2Then

xdxx 2)5(2

1 22

Substitute get)5( 2 xu Cx 32 5

6

1

xdxx 22 )5(

Note: This is exactly in the form! duu 2

21

Integrate this form to get Cu

32

1 3

Simplifying to get Cu

6

3

Note: If )5( 2 xu

Integrate

92 33 5x x dx3 2pick +5, then 3 u x du x dx

10

10

uC

9u du 103 5

10

xC

Sub to get Integrate Back Substitute

Ex. Evaluate 25 7x x dx

3/ 225 7

15

xC

Sub in

3/ 21

10 3/ 2

uC

Integrate

Pick u, compute du75 2 xu xdxdu 10

Sub inxdxxxdxx 10)75(10

1)75( 2

12212

9

Trig Integrals in the form:duun xdxx cossin 2

xxu sinsin Let Then xdxdu cosNote: This is exactly in the form! duu 2

Integrate this form to get Cu

3

3

Sub inCx

Cx

3

sin

3

)(sin 33

10

Basic Trig Integrals

2

2

cos sin

sin cos

sec tan

csc cot

sec tan sec

csc cot csc

u du u C

u du u C

u du u C

u du u C

u u du u C

u u du u C

The key to each basic Trig Integral is that:

Let u = The angle

While du = The derivative of the angle

First make sure you do not have a problem.

duun

You need to know the 6 trig. Derivatives, so that you can work backwards and find their Anti-derivatives!

12

Using the Trig Integrals

• The technique is often to find a u which is the angle, the argument of the trig function

• Consider

• What is the u, the du?

• Substitute, integrate

cos3x dx3 3u x du dx

1cos sin

31

sin 33

u du u C

x C

2 3x cosx dx Let u = x3 ; du = 3x2dx ; C.F. 1/3

3 21cosx (3x dx)

3

1cosudu

3

1sinu C

3

31sinx C

3

Symmetry in Definite Integral

Integrals of Symmetric Functions