chapter 3: advanced integration (part 2)

7/29/2019 CHAPTER 3: ADVANCED INTEGRATION (PART 2)

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NTEGRATION BY PARTS

1) Solve the integral using integration by partsmethod

Solution:

Step 1: Determine u and dv , their derivative and

integration.

u =x2 = sinx dxdu = 2xdx v = - cosx

Step 2: Arrange the equation into correct order as below

=

Step 3: If still in form of two functions, repeat theStep 1 again.

=

u1= 2x = cosx dxdu1 = 2 dx v1 = sinx

Step 4: Arrange the equation properly and solve the

integration part.

= = [ ]=

2) Solve the integral using integration by partsmethod

Step 1:

u = lnx =x2 dxdu = dx v =

Step 2: =

=

=

=

*

9 +

3) Solve the integral using integration by parts method

Step 1:

u = ex = sinx dxdu = exdx v = - cosx

Step 2: = - - = -

Step 3: -

u1= = cosx dxdu1 = dx v1 = sinx

Step 4: = =

2I ; :

EXERCISE

Solve the following integrals using integration by parts:

a)

b) 3 5c) 4

u = ? = ?du = ? v = ?


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NTEGRATION USING PARTIAL FRACTION

There are 4 types of partial fraction:o Linear functiono Repeated factor functiono Quadratic term functiono Improper fraction

1) Integrate :;: using partial fraction

Solution:

Step 1: Separate into partial fraction

3

3

Step 2: Multiply the equation

3 3

3 3 Step 3: Solve for the unknowns

x = 0 x = 1 x = -3

When x = 0,

0 + 1 =A (0 + 03) + 0 + 0

A = When x = 1,

(

3) 3 3

B = When x = -3,

3 ( 3) 6 3 () 3

C= 6

Step 4: Complete the partial fraction & integrate one by one.

:;:

;

:

= 3

2) Integrate ;;:: using partial fraction

Solution:

Step 1:;;

::

:

:

:

Step 2:

5

5 4 4 3

Step 3: When x = 1,

(-1)2 + 25 =A(1) +B(0) + C(0)

A = -2

Compare power of x,

x2 1 =A +B

1 = (-2) +B

B = 3

x -2 = 4A + 3B + C

-2 = 4(-2) + 3(3) + C

C= -3

Step 4: ;;::

:

:

: 3 3

3) Integrate ;: using partial fraction

Step 1:

;:

; :

:

Step 2: x2 =A(x2 + 1) + (Bx+ C) (x2)

x2 =A(x2 + 1) +B(x22x) + C(x2)

Step 3: When x = 2,

4 =A(5) +B(0) + C(0)

A =

Compare power of x,

x2 1 = A +B

1 = +B

B =

x 0 = -2B + C

= 5 C=


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33

Step 4: ; :

:

45

5

:

:

;

4) Integrate ;:;8; using partial fraction

Step 1:

3 5 6

5 6 3 3

3

3

Step 2: 18 =A(x2-3x) +B(x-3) + C(x2)

Step 3: When x = 0, When x = 3,

B = -6 C= 2

Compare power of x,

x2 0 =A + C

A = -2

Step 4: *

;+

6 3

x - 2

x43x3 ddd

-2x + 6x -2x3 + 6x2 i

-18

EXERCISE

Solve the following integrals using partial fraction:

a) ;7;;

b) ;;;

c) :;;:

d) ;::

chapter 3: advanced integration (part 2)

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