INDEFINITE INTEGRAL

BY

RASYID LATHIF AMHUDO

(D100102007)

CIVIL DEPARTMENT ENGINEERING FACULTY

MUHAMMADIYAH UNIVERSITY OF SURAKARTA

2010

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PREFACE

By saying grace to Allah S.W.T of abundance of grace and guidance then the process of

making this paper can be resolved in a timely manner.

Prepared a paper on this integral aims to find deeper about indefinite integral So we are

more obvious in studying Integral. for that on this occasion authors express gratitude to the

honorable:

1. Rena's mother who has given the chance, direction, encouragement, and facilities

associated with the writing of this paper.

2. To friends, who have provided guidance and assistance in writing this paper.

3. Various parties can not mantion one by one author.

Hopefully all the good deeds that have been given, get a penalty from Almighty God. The

author realizes that this paper are still many shortcomings and is far from perfect. For that

advice and constructive criticism so I hoped.

Hopefully this paper can provide maximum benefits for writers and anyone who reads,

Amen.

Author

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TABLE OF CONTENTS

PREFACE ii

TABLE OF CONTENTS iii

I. NOTATION INTEGRAL & INTEGRATION INDEFINITE 1

II. INDEFINITE INTEGRAL IN ALGEBRA FUNCTIONS 2

A. INDEFINITE INTEGRAL FORMULAS THEOREM IN THE ALGEBRA 2

B. WORK ON INDEFINITE INTEGRAL IN ALGEBRA FUNCTIONS 2

III. INDEFINITE INTEGRAL IN TRIGONOMETRY FUNCTIONS 4

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A. THEOREM FORMULAS INDEFINITE INTEGRAL IN TRIGONOMETRY 5

B. WORK ON INDEFINITE INTEGRAL TRIGONOMETRY FUNCTIONS 6

IV. INTEGRATION WITH FORMULAS INTEGRAL SUBTITUTIONS 7

V. INTEGRATION WITH FORMULAS INTEGRAL PARSIAL 9

REFERENCES 10

CHAPTER I

NOTATION INTEGRAL & INTEGRATION INDEFINITE

If n is any rational number except (-1), then:

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Integration function of the variable x is written

in named as the integral form of indeterminate function of x. Indefinite

integral of a function of x is as common functions that are determined by the relationship

By definitions:

a. F (x) called a general integral function and F (x) is F '(x) = f (x)

b. Integral Not Sure often called the Anti-derivatives

c. In a notation called the integral sign

d. . f (x) called the integrands

e. Arbitrary real constants C and is often called the integration constants

CHAPTER II

INDEFINITE INTEGRAL IN ALGEBRA FUNCTIONS

Indefinite Integral Formulas Theorem In The Algebra

1. (i) ∫ dx = x + C

(ii) ∫ a dx = ax + C

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2. (i) ∫{ f(x) + g(x)} dx = ∫ f(x) dx + ∫ g(x) dx

(ii) ∫{ f(x) - g(x) } dx = ∫ f(x) dx - ∫ g(x) dx

3. (i) ∫ xn dx = , with rational numbers n and n ≠ -1

(ii) ∫ axn dx = , with rational numbers n and n ≠ -1

Work On Indefinite Integral In Algebra Functions

Example!

Find indefinite integral in this algebra functions!

1. ∫ 5x4 dx 4.

2.

3.

Answer:

a. ∫ 5x4 dx =

So indefinite integral from ∫ 5x4 dx = x5 + C

b.

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So indefinite integral from

c.

So indefinite integral from

d.

So indefinite integral from

CHAPTER III

INDEFINITE INTEGRAL IN TRIGONOMETRY FUNCTIONS

in defining and designing the rules of indefinite integrals of trigonometric

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No F(x) F’(x)=f(x)

1.

2.

3.

4.

5.

6.

Sin x

Cos x

Tan x

Cot x

Sec x

Cosec x

Cos x

- Sin x

Sec2 x

- Cosec2 x

Tan x . Sec x

- Cot x . Cosec x

functions, it must be remembered is a derivative of trigonometric functions, as shown in table 1.2.

Table 1.2.

by using rules that have indfinite integral ∫ f(x) dx = F(x) + C indeterminate nature that

F’(x)= f(x) and derivative of trigonometry functions in the table 1.2., then the indfinite integral of trigonometry functions can be formulated as follows.

Theorem Formulas Indefinite Integral In Trigonometry

1. ∫ Cos x dx = Sin x + C

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2. ∫ Sin x dx = - Cos x + C

3. ∫ Sec2 x dx = Tan x + C

4. ∫ Cosec2 x dx = - Cot x + C

5. ∫ Tan x . Sec x dx = Sec x + C

6. ∫ Cot x . Cosec x dx = - Cosec x + C

Theorem Formulas Indefinite Integral In Trigonometry in the rule variable angle ax+b

1. ∫ Cos (ax + b) dx = Sin (ax + b) + C

2. ∫ Sin (ax + b) dx = - Cos (ax + b) + C

3. ∫ Sec2 (ax + b) dx = Tan (ax + b) + C

4. ∫ Cosec2 (ax + b) dx = - Cot (ax + b) + C

5. ∫ Tan (ax + b) . Sec (ax + b) dx = Sec (ax + b) + C

6. ∫ Cot (ax + b) . Cosec (ax + b) dx = - Cosec (ax + b) + C

where a and b is a real number with a≠0

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Work On Indefinite Integral Trigonometry Functions

Example!

Find indefinite integral in this trigonometry functions!

a. ∫ (x2 + sin x) dx

b. ∫ (sin x – cos x) dx

c. ∫ 2 sec2 x dx

d. ∫ (sec2 x – tan x. Sec x) dx

e. ∫ (tan2 x + 4) dx

Answer:

a. ∫ (x2 + sin x) dx = ∫ x2 dx + ∫ sin x dx =

b. ∫ (sin x – cos x) dx = ∫ sin x dx - ∫ cos dx = - cos x -sin x + c

c. ∫ 2 sec2 x dx = 2 ∫ sec2 x dx = 2 tan x + c

d. ∫ (sec2 x – tan x. Sec x) dx = ∫ sec2 x dx - ∫ tan x. Sec x dx

= tan x – sec x + c

e. ∫ (tan2 x + 4) dx = ∫ ((tan2 x +1)+3) dx = ∫(sec2 x + 3)

= ∫ sec2 x dx + ∫ 3 dx = tan x + 3x + c

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BAB IV

INTEGRATION WITH FORMULAS INTEGRAL SUBTITUTIONS

in solving the integral requires special techniques, one of which is to use the integral formula

substitution, there are many kinds of integral substitution formula that is used in everyday

situations, namely:

Integrations that can the change to deep shape ∫ f(u) du

Example !

Find this integral functions:

a. ∫ (2x + 5)9 dx

b. ∫ (px + q)n dx

c. ∫ dx

Answer:

a. ∫ (2x + 5)9 dx if u= 2x+5, then , or dx = du ,

Subtitution 2x + 5 = u & dx = du, then ∫ (2x + 5)9 dx can be modified Being

∫ u9 ( du) = ∫ u9 du

∫ u9 du =

so, ∫ (2x + 5)9 dx =

b. ∫ (px + q)n dx if u= px + q,than ,or dx = du ,

then ∫ (px + q)n dx can be modified Being ∫ un du

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Jadi, ∫ (px + q)n dx =

c. ∫ dx, Misalkan u = x2 + 2x + 6

maka ,du= (2x + 2) dx,atau (x + 1) dx = ½ du

∫ dx =

∫ dx =

=

=

Jadi, ∫ dx =

BAB IV

INTEGRATIONS WITH PARTIAL INTEGRAL FORMULAS

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Suppose u (x) and v (x) each are a function of the variable x, then

∫ u dv integration is determined by the relationship:

The relationship above shows that the integration is ∫ u dv can be converted to ∫ v du

integration, and vice versa.Success or failure of integration using partial integral formula is

determined by two things:

1. 1.Memilih part dv so that v can immediately be determined through the relationship v = ∫

dv

2. ∫ v du should be more easily resolved than with ∫ u dv

Contoh !

1.Dengan using partial integral formula, find ∫ x sin x dx.

      Suppose u = x, and dv = sin x dx, the integral ∫ x sin x dx can be written as:

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∫ x sin x dx = u v - ∫ v du

∫ x sin x dx = x (-cos x) - ∫ (-cos x) dx

= -x cos x + ∫ cos x dx

= -x cos x + sin x + C

1. ∫ x sin (x – π) dx = u v - ∫ v du

= x (- cos (x- π)) - ∫ (- cos (x- π)) dx

= - x cos (x- π)+ sin (x- π)

So the partial integral ∫ x sin (x - π) dx is = -x cos (x- π)+ sin (x- π)

2. = u v - ∫ v du

=

=

=

Jadi integral parsial adalah =

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REFERENCES

Purcell, Edwin J., Calculus with Analytic Geometriy (Fith Edition), Prentice-Hall

International Inc., Englewood Cliffts, 1987.

Sartono Wirodikromo, 2000 for high school Mathematics vol 1 to 6, Erland, Jakarta 2003.

www.google.com

www.itb.ac.id

www.stttelkom.ac.id

www.umslibrary.com

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