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(1)

The Integral

Indefinite Integral………...………….…2

Integration by Substitution………..…4

Definite Integral………...7

(2)

Indefinite Integral

An antiderivative of the function f is a function F so that

( ) ( ),

F x′ = f x ∀ ∈x I

Example: ( ) 1 3 3

F x = x + C is antiderivative of f (x) = x2

because of . F x′( ) = f x( ) because of . F x( ) = f x( )

The antiderivative of the function f is not unique, but they are different by constant.

The collection of antiderivatives of function f is called the indefinite integral of f with respect to x and denote by

( ) ( )

(3)

1

1. , 1

1

2. sin cos

r

r x

x dx C r

r

x dx x C

+

= + ≠ −

+

= − +

Some of Integral

Formulas

2 2

3. cos sin

4. sec tan

5. csc cot

x dx x C

x dx x C

x dx x C

= +

= +

(4)

Integration

by Substitution

Let , and F antiderivative of f , then

( ),

( )

u

=

g x

du

=

g x dx

( ( )) ( ) ( ) ( ) ( ( ))

f g x g x dx′ = f u du = F u + C = F g x + C

Example: Example:

( )

sin 2x +1 dx

Answer: Let 2 1, 2 1

2 u = x + du = dxdx = du

(

)

(

)

1

sin 2 1 sin 2

1 1

cos cos 2 1

2 2

x dx u du

u C x C

+ =

= − + = − + +

(5)

2. Find

3 2

2 1 1, 3

3

u x du x dx dx du

x

= + = → =

then ( 3 1)10 5 10 5 2 1 10 3

3 3

du

x x dx u x u x du x

+ = =

+ x dx

x3 1)10 5

(

Answer: Let

chapter

6

The Integral

because then , thus u = x3 +1

x

3

=

u

1

3 10 5 1 10 1 11 10

3 3

12 11

1 1

36 33

3 12 3 11

1 1

36 33

( 1) ( 1)

( 1) ( 1)

x x dx u u du u u du

u u C

x x C

+ = − = −

= − +

(6)

1 2

Sigma Notation

(7)

b definite integral as limit of Riemann sum.

Definite Integral

b

(8)

a x2 xk1 xk b

4. Defined Riemann sum

=

The Integral

k

(9)

Use Riemann sum to evaluate

2

0

(x−2)dx

Answer:

(i) Divide interval [0,2] into n subinterval all of the same length, that is

Example

n

x = 2

x0 = 0

n x x1 = 0+∆ = 2

n .

x

x2 = 0+2∆ = 22

n i

i i x

x = 0+ ∆ = 2

0 2

x

∆ ∆x

x

x

1

(10)

(ii) Choose ci = xi

(iii) Defined Riemann sum

( )

(

)

(

2

)

2 2 4 4

2

1 1 1 1 1

4 4

2 1

n n n n n

i i

i i n n n n

i i i i i

f c x i

n n

= = = = =

∆ = − = − = −

chapter

6

The Integral

1 1 1 1 1

2

4 ( 1) 4 2

2 2

i i i n i n i

n n

n

n n n

= = = = =

+

= − = − +

(iv) If thenn →∞

(

)

2

2

0

( 2) lim 2 n 2

n

x dx

→∞

(11)

Remark: If function y = f(x) positive on interval [a,b] then definite integral = area of region under the graph of y = f(x) over the interval [a,b].

Some properties of definite integral

chapter

6

The Integral

[

p f x q g x dx

]

p f x dx q g x dx

a b

a b

a b

( ) + ( ) = ( ) + ( ) 1. Linear

2. If a < b < c, then

f x dx f x dx f x dx

a c

a b

b c

(12)

f x dx

The Integral

5. If f(x) is an even function, then f x dx f x dx

Example: Evaluate

Answer:

4 2 4 2

(13)

The Fundamental

Theorem of Calculus Part 1

If f is continuous on [a,b] then is( ) ( )

x

a

F x = f t dt

continuous on [a,b] and differentiable and its

The Fundamental Theorem of Calculus Part 1

continuous on [a,b] and differentiable and its

( ) ( ) ( )

x

a

d

F x f t dt f x

dx

′ = =

(14)

( )

( ) ( ( )) ( )

u x

d

f t dt f u x u x

dx = ′

From the fundamental theorem of calculus part 1, we can derive:

chapter

6

The Integral

a

dx

( )

( )

( ) ( ( )) ( ) ( ( )) ( )

v x u x

d

f t dt f v x v x f u x u x

(15)

+

The Integral

(16)

The Fundamental

Theorem of Calculus Part 2

If f is continuous at every point of [a,b] and F is

any antiderivative of f on [a,b] then

The Fundamental Theorem of Calculus Part 2

any antiderivative of f on [a,b] then

( ) ( ) ( )

b

a

(17)

( )

Example

(18)

The Integral

(19)

For problems 1-5, find the antiderivative F(x) + C of f(x).

5 10

3 )

(x = x2 + x+ f

) 6 7

20 ( )

(x = x2 x7 − x5 + f

1.

2.

Problem Set 1

f x

x x

( ) = 1 + 6

3 7

f x x x

x

( ) = 2 − 3 +1

3 2

2

3 4

( )

f x = x

2.

3.

4.

(20)

(

x2 − 4 2

)

3 x dx

(

x2 − 3x + 2

)

2

(

2x3

)

dx

6.

7.

For problems 1-5, evaluate each integral given.

Problem Set 2

3x 3x2 + 7 dx

(

5x2 +1 5

)

x3+ 3x −2 dx

3

2 2 5

y

y

dy +

(

cos42x

)

(

− 2 sin 2x dx

)

8.

9.

10.

(21)

For problem 12 - 15, evaluate .f x dx( )

0 5

f x x x

x x

( ) ,

,

= + ≤ <

− ≤ ≤

2 0 2

6 2 5

x , 0≤ x <1

12.

Problem Set 3

f x

x x

x

x x

( )

, ,

,

=

≤ <

≤ ≤ − < ≤

0 1

1 1 3

4 3 5

13.

14. f(x) = |x -1|

3 1 3

4

2 )

(x x x

f = −

(22)

3 2 3 1

For problem 16 - 21, evaluate each integral given.

8 7 2 2

sin cos

(23)
(24)

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