Part 8 (PDF)

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Part 8 (PDF)

Exercise 8

a) Find the indefinite integral for following functions.

a)

2 14 2

8 4 4 2

8 5 2 2

b)

4

2 3

2 1;

:


2 3;


2;


2

2;


2


4

2 3 1 2 1 2 · 4


1 2 2 1 2 2 2 3

1


c)

1 3

Integration by Parts:

· ·

;


1 3 ; 1

· 1 3 · 1 · 1

3

· 1 3 1 3

2


2. Find the definite integral for following functions.

a)

2

1. The Indefinite Integration by Substitution

: 2 1;


2;

2

2 1 2 2 1

22 2 1

22 2


2


1


22 2 1

22 2 2 1 · 60 86,56

2

2. The definite Integration by Substitution


2


2 1

1 3

3 7


1

22 2 1

22 2 2 86,56

3


)

3 ·

Integration by Parts:

· ·

;

3 ; 6

3 · · 3 · 6 · 3 · 6 · 6

3 2 2


3 ·


3 2 2

3 1 21 2 3 4 24 2 4,09

4


c)

4 5

The definite Integration by Substitution

5 ;


10;


10

1 5 · 1

5

3 5 · 3 45


4 5


4




4






1

10

1 10

4 · 1

4 · 1

10 3 3 1


10 45 1 5 79,81

10

5


3. Compute the total area between the follow functions and the -

axis and the lines and

a)

3 3 18

Between 3 1

The roots of the function:

3 3 18 3 6 0

6 0

1 √25

,

2

2; 3



3 3 18



3 3 18


| 1,5 18 | 1,5

18

|8 6 36 27 13,5 54| 1 1,5 18

8 6 36 8,5 40,5 49

6


)

1

Between 0,5 4

The roots of the function:

1

0; 0 1


The definite Integration by Substitution




2

2




,

1 1






2

2 ,




1 2 ln 1 1 2 ln 0,5 1 2 ln 4 1 2 ln1

0 1 2 ln0,5 1 2 ln4 0 1,20

7


c)

4 16

Between 1 1

The roots of the function:

4 16 0

2; 0; 2


4 16


1

2 16



4 16


1 2 16




0 0 1 2 16 1 16 0 0 31

2

8


4. Compute the total area between the follow curves and

:

a)

Intersections:

2 1;

2 1 1

1

2 1 2 1

4 0

4 0

0; 4

2 1: ; 1 :

2 1 1 2 1 1




2 1 2 1




1 3 · 4 2 · 4 0 0 10 2 3




4


1

3 2


9


)

Intersections:

2 ;


2


2 ; 0; 2

: ; 2 :





2 2

1






1


1 0 0 4 0 4 1 0 2 2 0,43



10

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