การหาปริพันธ์.pdf

E-learning 25 -02-55

1

( Indenfinite Integral ) ( )y f x dy

dx

dydx

y x 2dy x

dx

2y x 2 3y x 2y x 2dy x

dx 2y x c c

( )dy f xdx

( )y F x c

( )dy f xdx

( integration ) ( )y F x c

(solution) ( antiderivative ) ( )dy f xdx

( ) ( )F x f x ( )F x (antiderivative) ( )f x

2

2

2

2

3

( ) ;

xx

F x x

x c c

( )F x 2x 2x 2x c c


2

( ) ( )F x f x ( )F x c

(indefinite integral) f ( )f x dx ( ) ( )F x f x ( ) ( )f x dx F x c

( )f x ( integrand ) x (variable of integration) c

( )f x dx ( ) f x

(1) ( ) f x dx ( )f x (2) ( )dy f x

dx

( ) ( ) dy dx f x dxdx

( ) y f x dx

(3) ( )y F x c Y c 2y x c

2y x c 0 0( , )x y ( )y F x c


3

1 F f 2( ) 3 6 3f x x x (0) 1F F [0, 2] x c ( )F c

2( ) 3 6 3f x x x F f ( ) ( )F x f x 3 2 1( ) 3 3F x x x x c 1c 1(0) F c (0) 1F 1 1c 3 2( ) 3 3 1F x x x x (*) F [0, 2] x c 2 2 2( ) 3 6 3 3( 2 1) 3( 1)F x x x x x x 1x (0, 2) (*) ( )F x 0 , 1x 2 (0) 1 , (1) 0F F (2) 1F 1 ( ) 1F c #

k c (1) dx x c (2) kkdx x c

(3) 1

, 11

nn xx dx c n

n

(4) ( ) ( )kf x dx k f x dx (5) [ ( ) ( )] ( ) ( )f x g x dx f x dx g x dx


4

2 4 2 5 3 4dy x x xdx

(1) ( 1)y y (0)y (Ent. . 2541) 1. 0 2. 1

3. 2 4. 3

dydx

4 2 5 3 4x x x (1) ( 1)y y y 4 2 (5 3 4 )x x x dx 5 3 2 2x x x c c (1)y c ( 1)y 4 c (1)y ( 1)y c 4 c c 2 y 5 3 2 2 2x x x (0)y 2 # 3 ( ) ( )f x g x ( ) ( )g x h x (Ent. 2 1/2542) 1. ( ) ( ) ( )g x d x f x c 2. ( ) ( ) ( )h x d x f x c 3. ( ) ( ) ( )g x d x h x c 4. ( ) ( ) ( )f x d x f x c 1. ( ) ( ) ( ) ( ) ( )g x d x f x d x f x c 2. ( ) ( ) ( ) ( ) ( ) ( )h x d x g x d x g x c f x c 3. ( ) ( ) ( )g x d x g x c 4. ( ) ( ) ( )f x d x f x c

3. #


5

( ) f [ , ]a b ( ) ( )F x f x [ , ]x a b ( ) [ ( )] ( ) ( )

bb

a af x dx F x F b F a

( )y F x c ( ) 0 ( ) ( )y F x F x f x ( ) ( )f x dx F x c ( ) ( )f x dx F x c ( ) ( ) ( )

b

af x dx F b F a

(1) [ ( ) bakf x dx ( ) ,

b

ak f x dx k

(2) [ ( ) ( )] baf x g x dx ( ) ( ) ( )

b b

a af x d x g x dx

(3) ( ) baf x dx ( )

a

bf x dx

(4) ( ) aaf x dx 0

(5) [ , ]c a b ( ) ( ) ( ) b c ba a cf x dx f x dx f x dx

(6) ( ) 0f x [ , ]x a b ( x ) ( ) 0b

af x dx

(7) ( ) 0f x [ , ]x a b ( x ) ( ) 0b

af x dx

( ) f x dx x ( ) b

af x dx


6

1 42 1 2

21 0

1 (4 ) x dx x dxx

(Ent. . 2540) 1. 10 2. 14 3. 20 4. 24

42 1 2

21 0

1 (4 )x dx x dxx

12 12 2 2

1 0 ( ) (16 8 ) x x dx x x dx

12 33 2

2

1 0

1 16 163 3 2x xx x

x

8 1 1 16 1 1 16 03 2 3 3 2

14 # 2 ( )y f x ( , )x y 2 3 2x x

20

( ) 4f x dx ( , )x y ( )y f x (Ent 1/2546 1) 1. 40,

3

2. 40,3

3. 131,4

4. 131,4

( )y f x ( , )x y 2 3 2x x 2( ) 3 2f x x x 2( ) 3 2 f x x x dx

3 23 23 2x x x c c

20

( ) 4f x dx

3 22

0

3 2 43 2x x x c dx

24 3

2

0

412 2x x x cx

4 3

22 2 2 2 0 412 2

c


7

42 43

c

4 3

c

3 23 4( ) 2

3 2 3x xf x x

4(0) 3

f #

( )y f x [ , ]a b ( )y f x X x a x b

A ( ) 0f x ( ) 0f x X X ( )y f x X x a x b

1 ( )p

aA f x dx 2 ( )

q

pA f x dx

3 ( )r

qA f x dx 4 ( )

b

rA f x dx

1 2 3 4A A A A A

( )y f x

(1)

(3)

( ) ba

A f x dx


8

( )y f x x ,a b

* ( )x g y ,c d ( )x g y Y ,c d ( )x g y y y c y d

( ) dc

A g y dy 1 3( ) 4f x x x X 1,2

3( ) 4f x x x X 3 4 0x x ( 2) ( 2) 0x x x 0, 2, 2x X 0 , 2 , 2x x x 0 1, 2 A 0 23 3

1 0 ( 4 ) ( 4 )x x dx x x dx

0 24 4

2 2

1 0

2 24 4x xx x

1 0 2 (4 8) 04

7 23 44 4

5.75 #

f ,a b ( )y f x X x a x b (1) x ,a b (2) x (3)


9

2 3 3 2y x x 0x 2x X (Ent. 1 1/2541) 1. 3

2 2. 1

6

3. 23

4. 56

2 3 2y x x X 2 3 2 0x x ( 2)( 1) 0x x X 2x 1x

X 0x 1x

13 21 2

00

3 3 2 23 2x xx x dx x

1 3 2 03 2

5 6

# 2 3 2y x x 0y 0,1x X 1 2

03 2 x x dx


10

3 2( )f x x c c 4c ( )y f x 2x 1x 24 c ( Ent. 1 2/2543) 2 ( ) , 4y f x x c c ( ) ( ) ( )f x x c x c ( )y f x X x c x c [ 2 , 1] [ , ]c c ( )y f x 2x 1x 24 1 2

2( ) 24x c dx

13

2

243x cx

1 8 2 243 3

c c

(3 3 ) 24c 3 3 24c 3 27c 9c #

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