Kalkulus Integral

 





´

 

´

n

´ ´

xn dx!

1



xn 1  c

n 1

n

{1

n

cx dx ! c x dx 





1

7



´

x dx!



´

x dx!



´

9 x ! 9.

7 1 1

8

x81  c!

8 1 1

2

1

x7 1  c!

8 1

x9  c

9



2 1

x8  c

x2 1  c !

9

x3  c ! 3 x3  c

3

dx ! cx  c ´ c dx



´

´

´

3 3   (4 x  6) dx! 4 x dx 6 dx 

! 4.

  

!

4 4

1

3 1

1

31

x  6.

0 1

0 1

x  c

4

x  6 x c

! x4  6 x c



´ k  x dx ! k ´ x n

n

c

 

´

6 x2 dx ! 6

´

x2 dx  c !

6 2 1



x2 1  c !

6 3

3

3

x  c! 2 x  c



´ k d  x ! k  x  c 



´  ´ 4 d  x! 4 x c  3 d x! 3 x c



´ (a x  n

´a x ´ b x 

bxn )d  x !

n

n

c

 

´ ( x  8 x ) dx ! ´ x 6

3

dx

6

1

 !

1 7

3

6

x 1  c 8.

6 1

!

´ 8 x dx 1

31

3

x 1 c

x7  2 x4  c

 

´ (5 x  2) dx! ´ 5 4

!5

x4 

1

´ 2 dx

c

x5  2 x c

5 5

! x  2 x c !6



1

1



x1 2  4

1 2

11



x1 1  3 x c

´ (6 x  4 x 3) dx! ´ 6 x dx ´ 4 x dx´ 3 dx 2

2

 6 !



3

3

x 

4

x2  3 x c

2

! 2 x3  2 x  3 x c 2

 

´ (2 x 8 x ) dx! ´ 2 x dx´ 8 x dx c 3

3

!2 !

1 11

2

x2 

11

x 8

8

4

2

1 3

x c

4 ! x  2 x4  c 2

1

1 3

x  c

c



´ ( x  3) dx! ´



!





´

x5 dx 3 dx

5

!

1

1

 x 5

5 1

1

6

d x  3 x 

c

x dx  3 x  c 6

´ 9 x d  x  ´ 6 x d  x ! 9 ´ x d  x  6 ´ x d  x 2

2

3

3

 ! 9 1 x21  c 6 1 x31  c  2 1 3 1  9 3 6 4 x c x c !    3 4 3 4 3 ! 3 x  c x  c

!3 x  3



2

3

x  c 4

2

 g ( x) 

r  {  1 

´  g ( x )r . g '( x)d  x !

1

r   1

g ( x )

 g '( x)  r 1

 c 

 g ( x) 

 g ( x) 

 g '( x) 

  

´

3

1

4

( x  2) 3x 2 d  x!



4 1

3

( x 2)

1

4 1

 c

! ( x3  2)  c 5

5

1



´

( x  3 x)2 (4 x  3)d  x !



´

3 (6 x 2) (3 x2  2 x) dx!



´

3 4 (8 x  1) (2 x  x) dx !

4

3

´

  si  sin x.cos x dx!  !

1 1 1 1 2

4

2 1

1

31 1

11

1

4  (2 x  x)1 1 !

4

3

3 (3 x2  2 x) 1 !

 (sin x)1 1  c

(sin  x )2  c



( x  3 x)2 1 !

1 2

3

( x  3 x)  c 1

4

4 (3 x2  2 x)  c

4 (2 x  x) 2  c

 d  d  x d  d  x d  d  x d  d  x d  d  x d  d  x d 

n

 x ! n x

 x n

n 1

´ xn d  x ! n  1  c ´ cos x d  x! sin

sin  x ! cos x cos x !  sin

tan  x

´ sec

2

2

x c

´ sin  x d  x! cos x c

x

! s ec x

cot  x ! csc

1

2

x dx ! tan x c

´ csc  x d  x ! cot 2

x

x c

sec

x! sec xtan x

´ sec xtan x dx! sec x c

csc

x! csc xco cot x

´ csc x cot x d  x! csc x c

dx 2

2 ¨ 1 ¸ ¨ sin x ¸ © ¹ © ¹ dx ! sec x  tan x  dx ª cos º ª cos x º



2

! sec  x

 tan x 

d  x

 b

´ f ( x) dx !

f ( b)  f ( a) p  

f  

a

 1

´

  x 2 d  x ! 0

 !  !  !

1

3 1

3 1

3 1

3

1

x 3 A0  3

1 

0

1

3

0

3



 f ( x ) 

2



2

2

´ 2 x  3 d  x ! ´ 2 x d  x  ´ 3d  x 1

1

1

1

1

2

 ! 2  x 2 3x A1  2

2

 !  x 2  3 x A1   ! (22  3.2)  (12  3.1)  ! (4  6)  (1  3)  ! 10  4  ! 6

 1



´(

x2  2)

2

x dx

0 2

 g ( x ) ! x  2  g '( x ) ! 2 x 1 2

1

( x 2  2) 2 A0

! !

1 2

1 2

(12  2) 2  .9 

1

1 2

(02  2) 2

.4

2

! 4, 5  2 ! 2, 5  1



´(

x2  2)

2

x dx

0



2

! x 2

u

d u

! 2 x p d u ! 2 x d  x

d  x



 x ! 0 p u ! 02  2 !

2

 x ! 1 p u ! 12  2 ! 3 3

´

u

d u !

u

2

2

2

!

1

1 2

.

32 

1 2

3

A2 2

.2 !

1 2

.9 

1 2

.

4 ! 4, 5  2 ! 2, 5

 2



2

´ (3 x

2

2

´

3

´

2

3

 4 x ) d  x ! 3 x d  x  4 x d  x

1

1

1

1

1 2 3    x 2 1  4 x 1 A1 !3 3 1  2  1 2 3 4  !  x  x A1  3 4 3 4 ! ( 2  2 )  (1  1 )  ! (8  16)  (1  1)  ! 24  2  ! 22 1



1

1

´ (6 x  4) dx! ´ 6 x dx ´ 4 2

2

0

0

dx

0

1 1  3  x 1  4 x A0 !6  2  1 1 4  ! 2 x  4 x A0 3 3  ! ( 2.1  4.1)  ( 2.0  4.0)  ! (2  4)  (0  0) !6 1



´ ( x  2 x  3 x 2) 3

2

2

dx (3 x2  4 x 3) dx

0



! x 3 2 x  3 x  2 2

u

d u dx



! 3 x  4 x  3  2 2

 x ! 0 p u ! 03  2.0 2  3.0  2 !

2

 x ! 1 p u ! 1 3  2 .1 2  3.1  2 !

8

8

´

u

2

d u !

1 u

3

3

2

! ! ! !

1

3

.8

3 1

3

8

A2 

1

3

. 5 12 

5 12

3 5 04

3 ! 16 8



3

.2

8

3

1

3

.8

2



´ ( x  2 x  5 x) (4 x  6 x  5) d  x 4

3

3

2

1



! x 4  2 x3  5 x

u

d u dx

! 4 x 3  6 x  5 2

 x ! 1 p u ! 14  2 .13  5 .1 ! 8



 x ! 2 p u ! 24  2 . 23  5 . 2 ! 42 42

´

d u !

u

8

! ! ! !

1 u

2 1

2

1 2

2

42

A8

1

422 

2

1764 

1764 2



2

8

1 2

64

64 2

1700 2

! 850 T 

2



´ T 



2 (sin )x cos 2

! sin x

u

d u dx



x dx

! cos xdx

 x ! T

2

T

 x !

2

p u ! sin(T ) !  sin T  2

p u ! sin T 

1

´

u

2

d u !

1 u

! ! ! !

1

3 1

A1

3 1

3 2

3

3

1 

 

!1

1

3

3

1

2

1

3

¨ 1¸ © 3¹ ª º 1

3

3

1

2

! 1











n

§ i ! 0  1  2  3  ............ (n  1) n

S  !

p  

i !0

n

§a

k

0

1

2

! a  a  a ............ a

n

S  !

p  

k ! 0

n

§i

2

2

2

2

! 0  1  2 ............  n

2

i !0



 4



§ 3. i ! 3.2  3.3  3.4 i !2

! 6  9  12 ! 27

3



§i

3

i !1

3

3

3

!1  2 3

! 1  8  27 ! 36

4



§i

2

2

2

2

 1 ! ( 2  1)  (3  1)  (4  1)

i !2

! (4  1)  (9  1)  (16  1) ! 3  8  15 ! 26

4



§i

2

2

i !2

§ (2i i !1

2

! (4  2)  (9  3)  (16  4) ! 2  6  12 ! 20

3



2

 i ! ( 2  2)  (3  3)  ( 4  4)

2

 1) ! (2.1  1)  (2.2  2)  (2.3  3) 2

2

2

2

! (3) (3)  (10)  ( 21) ! 9  100  441 ! 550 2

2

2

2

2

2

n . ( n  1) 2

1  a



n 1

a 1





 

´ ( x

  4

  

´

 

´

4

 3) d  x ! x d  x  3 d  x

!

1 5

x5  3 x c

1





1

1

´ (3 x  5 x ) dx! 3´ x dx5´ x dx 2

4

4

2

0

0

0

1

3

! 3  x  5

3

1 5

x

1

5

A0

1

! ( x  x ) A0 3

5

3

3

5

5

! (1  1 )  (0  0 ) ! (1  1)  (0  0) ! 20 !2



´

3 (4 x  2 x2  1) (12 x2  4 x) d  x!

T 



´

(tan  x)(sec 2 x) d  x !

0

! !

1 2

1 2

1

1 2

3 (4 x  2 x2  1) 2  c

T 

(tan x) 2 A0 2

(tan  x )  .0

2

2



1

.0

1 2

(tan 0)

2

2

2

!0 3



§ (2i  5) i !1

2

2

2

! ( 2.1  5)  ( 2.2  5)  ( 2.3  5)

! (7)  (9)  (11) ! 49  81  121 ! 251 2

2

2

2

p

 g ( x)