math 1431 dr. melahat almus almus ...almus/1431_day38_after.pdf · if you e-mail me, please mention...

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Math 1431

Dr. Melahat Almus

almus@math.uh.edu

http://www.math.uh.edu/~almus

OFFICE HOURS (610 PGH)

MWF 9-9:45am, 11-11:45am

COURSE WEBSITE:

http://www.math.uh.edu/~almus/1431_sp16.html

Visit my website regularly for announcements and course material!

If you e-mail me, please mention your course (1431) in the subject line.

BUBBLE IN PS ID VERY CAREFULLY! If you make a bubbling mistake, your scantron will not be saved in the system and you will not get credit for it even if you turned it in.

Bubble in Popper Number.

Be considerate of others in class. Respect your friends and do not distract anyone during the lecture.

DID YOU RESERVE A SEAT FOR TEST 4?

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Chapter 6 – Integration

Section 6.2

Theorem: Fundamental Theorem of Calculus Part 1

If f is a continuous function over the interval a,b , then the function

x

a

F x f t dt

is continuous on a,b and differentiable on a,b . Moreover,

x

a

dF ' x f t dt f x

dx , for all x in a,b .

Theorem: Fundamental Theorem of Calculus Part 2

Let f be a continuous function over the interval a,b . If G is any antiderivavite for

f over the interval a,b , then

b

a

f x dx G b G a .

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Example: Calculate the definite integrals using FTOC.

2

4

/

/

cos x dx

1

4 3

0

5 4 6x x dx

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Section 6.3 – Basic Integration Rules

Indefinite integral:

The notation f x dx is used for an antiderivative of f and called an indefinite integral.

f x dx F x means F' x f x .

In general, to find f x dx , we find an antiderivative of f x , say F x , and then we write

the indefinite integral as:

f x dx F x C .

Here, C is called the constant of integration.

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Basic Rules of Integration

1) Power Rule for Integrals

1

1

rr x

x dx Cr

, where 1r .

Examples:

4x dx

x dx

6x dx

1dx

x

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2) k f x dx k f x dx , where k is a constant number.

3) f x g x dx f x dx g x dx .

4) k dx k x C , where k is a constant number.

Using these rules, we can integrate any polynomial.

Examples:

8 35 7x x x dx

5 3

3

5 4x x xdx

x

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2x xdx

x

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Basic Formulas

Using the derivative formulas you learned in the previous chapters, we can derive several formulas for integration.

1) Integrals of Basic Trigonometric Functions:

sin x dx cos x C

cos x dx sin x C

2sec x dx tan x C

2csc x dx cot x C

sec xtan x dx sec x C

csc xcot x dx csc x C

2) 1

dx ln x Cx

.

3) Integrals of Exponential Functions

x xe dx e C

xx a

a dx Cln a

, where 0 1a , a .

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4) Integrals Resulting in Inverse Trigonometric Functions

The following formulas are derived using the derivative formulas for inverse trigonometric functions (Chapter 4).

2

1

1dx arcsin x C

x

,

21

1dx arctan x C

x

,

2

1

1dx arc sec x C

x x

.

5) Integrals of Hyperbolic Functions

sinh x dx cosh x C

cosh x dx sinh x C

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TABLE OF INTEGRALS

1

1

rr x

x dx Cr

; 1r 1

dx ln x Cx

sin x dx cos x C cos x dx sin x C

2sec x dx tan x C 2csc x dx cot x C

sec xtan x dx sec x C csc xcot x dx csc x C

x xe dx e C x

x aa dx C

ln a ; 0 1a , a .

sinh x dx cosh x C cosh x dx sinh x C

2

1

1dx arcsin x C

x

21

1dx arctan x C

x

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Examples:

5sin 2cosx x dx

2

4

1dx

x

2

6

1dx

x

1x dx

x

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sinhxe x dx

22 secx x dx

2 2

1

2xdx

x