The document defines integration as the inverse operation of differentiation or the antiderivative. Integration finds the function given its derivative, while differentiation finds the derivative of a function. The key points are:
1) Integration is denoted by the integral sign โซ and finds the antiderivative F(x) of a function f(x) plus a constant c.
2) Some basic integration rules and theorems are presented, including formulas for integrating polynomials and trigonometric functions.
3) The substitution rule is described for performing integral substitutions to solve integrals that can't be solved with basic formulas. Examples of integrating trigonometric functions and expressions involving square roots are provided.
ICT Role in 21st Century Education & its Challenges.pptx
ย
Integral dalam Bahasa Inggris
1. A. Definition of Integration
In Class XI, you have learned the concept of derivative. Comprehension
on the derivative concept you can use to understand
integration concept. For that, try to determine the following derivative functions:
๏ท ๐1(๐ฅ) = 3๐ฅ3
+ 3
๏ท ๐2(๐ฅ) = 3๐ฅ3
+ 7
๏ท ๐3(๐ฅ) = 3๐ฅ3
โ 1
๏ท ๐4(๐ฅ) = 3๐ฅ3
โ 10
๏ท ๐5(๐ฅ) = 3๐ฅ3
โ 99
Note that these functions have the general form , ๐(๐ฅ) = 3๐ฅ3
+ ๐ with c is constant. Each
function has a derivative ๐โฒ(๐ฅ) = 9๐ฅ2
. Thus, the derivative function ๐(๐ฅ) = 3๐ฅ3
+ ๐ is ๐โฒ(๐ฅ) =
9๐ฅ2
.
Now,what if you have to define the function ๐(๐ฅ) of
๐โฒ(๐ฅ) is known?Determine the function ๐(๐ฅ) from ๐โฒ(๐ฅ), means determining
antiderivative of ๐โฒ(๐ฅ). Thus, the integration is the antiderivative
(Antidiferensial) or the inverse operation of the differential.
If ๐น(๐ฅ) is general function y that is common ๐นโฒ
(๐ฅ) = ๐(๐ฅ), then ๐น(๐ฅ) is antiderivative
or integral of ๐(๐ฅ).
Integration function ๐(๐ฅ) with respect to ๐ฅ is denoted as follows:
โซ ๐(๐ฅ) ๐๐ฅ = ๐น(๐ฅ) + ๐
With :
โซ = integration
๐(๐ฅ) = function integration
๐น(๐ฅ) = integration common function
๐ = constanta
Now, consider the derivative of the following functions ;
๐1(๐ฅ) = ๐ฅ, be obtained ๐1โฒ(๐ฅ) = 1
So, if ๐1โฒ(๐ฅ) = 1,then ๐1(๐ฅ) = โซ ๐1
โฒ (๐ฅ) ๐๐ฅ = ๐ฅ + ๐
๐2(๐ฅ) =
1
2
๐ฅ2
, be obtained ๐2โฒ(๐ฅ) = ๐ฅ
So,if ๐2โฒ(๐ฅ) = ๐ฅ, then ๐2(๐ฅ) = โซ ๐2
โฒ (๐ฅ) ๐๐ฅ =
1
2
๐ฅ2
+ ๐
๐3(๐ฅ) =
1
3
๐ฅ3
, be obtained ๐3โฒ(๐ฅ) = ๐ฅ
So,if ๐3โฒ(๐ฅ) = ๐ฅ, then ๐3(๐ฅ) = โซ ๐3 โฒ(๐ฅ) ๐๐ฅ =
1
3
๐ฅ3
+ ๐
2. ๐4(๐ฅ) =
1
6
๐ฅ6
,be obtained ๐4โฒ(๐ฅ) = ๐ฅ5
So,if ๐4โฒ(๐ฅ) = ๐ฅ5
, then ๐4(๐ฅ) = โซ ๐4 โฒ(๐ฅ) ๐๐ฅ =
1
6
๐ฅ6
+ ๐
Of this description, it appears that if ๐โฒ(๐ฅ) = ๐ฅ ๐
, then ๐(๐ฅ) =
1
๐+1
๐ฅ ๐+1
+ ๐ or
can be written โซ ๐ฅ ๐
๐๐ฅ =
1
๐+1
๐ฅ ๐+1
+ ๐, ๐ โ โ1.
For example, the derivative function ๐(๐ฅ) = 3๐ฅ3
+ ๐ is ๐โฒ(๐ฅ) = 9๐ฅ2
.
This means, antiderivative of ๐โฒ(๐ฅ) = 9๐ฅ2
is ๐(๐ฅ) = 3๐ฅ3
+ ๐ or written โซ ๐โฒ(๐ฅ)๐๐ฅ = 3๐ฅ2
+ ๐.
This description illustrates the following relationship.
If ๐โฒ(๐ฅ) = ๐ฅ ๐
,then ๐(๐ฅ) =
1
๐+1
๐ฅ ๐+1
+ ๐ , ๐ โ โ1, with c is a constant.
Example:
1. Find the derivative of each of the following functions :
Answers:
2. Find the antiderivative x if known:
Answers:
3. B. Indefinite Integrals
In the previous part, you have known that the integral is an antiderivative. So, if there
is a function F(x) that can differential at intervals [๐, ๐] so that
๐(๐น(๐ฅ))
๐๐ฅ
= ๐(๐ฅ),the
antiderivative of f (x) is F (x) + c.
Mathematically, written
โซ ๐(๐ฅ)๐๐ฅ = ๐น(๐ฅ) + ๐
where,โซ ๐๐ฅ = symbol of stated integral antiderivative operation
f(x) = integrand functions, namely functions which sought antiderivative
c = constant
For example, you can write
Because,
So you can look at indefinite integral as representatives of the whole family of functions (one
antiderivative for each value constant c. The definition can be used to prove
the following theorems which will help in the execution of arithmetic
integrals.
Theorem 1
If n is a rational number and n โ โ1,then โซ ๐ฅ ๐
๐๐ฅ =
1
๐+1
๐ฅ ๐+1
+ ๐ where
c is a constant.
Theorem 2
If f the integral function and k is a constant, then โซ k f(x) dx = k โซ f (x) dx
Theorem 3
If f and g is integral functions, then โซ (๐(๐ฅ) + ๐(๐ฅ))๐๐ฅ = โซ ๐(๐ฅ)๐๐ฅ + โซ ๐(๐ฅ)๐๐ฅ
Theorem 4
If f and g is integral functions, then โซ (๐(๐ฅ) โ ๐(๐ฅ))๐๐ฅ = โซ ๐(๐ฅ)๐๐ฅ โ โซ ๐(๐ฅ)๐๐ฅ
Theorem 5
Substitution Integrals Rule
If u is a function which can differential and r is a numbers which no zero, then
โซ (๐ข(๐ฅ))
๐
๐ขโฒ(๐ฅ)๐๐ฅ =
1
๐+1
(๐ข(๐ฅ))
๐+1
+ ๐, where c is a constant and rโ โ1
4. Theorem 6
Partial Integrals Rule
If u and v is a functions which can differential, thenโซ ๐ข ๐๐ฃ = ๐ข๐ฃ โ โซ๐ฃ ๐๐ข
Theorem 7
Trigonometri Integrals Rule
๏ท โซ ๐ ๐๐ ๐๐ฅ = โcos ๐ฅ + ๐
๏ท โซ ๐๐๐ ๐๐ฅ = sin ๐ฅ + ๐
๏ท โซ
1
๐๐๐ 2 ๐ฅ
๐๐ฅ = tan ๐ฅ + ๐
Where c is a constant
Prove Theorem 1
For prove theorem 1,we can differential ๐ฅ ๐+1
+ ๐ which be found at right space the
following ;
๐
๐๐ฅ
(๐ฅ ๐+1
+ ๐) = (๐ + 1)๐ฅ ๐
โฆ . ๐๐ข๐๐ก๐๐๐๐ฆ ๐ก๐ค๐ ๐ ๐๐๐๐ ๐ค๐๐กโ
1
๐ + 1
1
๐ + 1
.
๐
๐๐ฅ
(๐ฅ ๐+1
+ ๐) = (๐ + 1)๐ฅ ๐
.
1
๐ + 1
๐
๐๐ฅ
[
๐ฅ ๐+1
๐ + 1
+ ๐] = ๐ฅ ๐
So, โซ ๐ฅ ๐
๐๐ฅ =
1
๐+1
๐ฅ ๐+1
+ ๐
Prove Theorem 3 and 4
For prove theorem 4,we can differential โซ ๐(๐ฅ)๐๐ฅ ยฑ โซ ๐(๐ฅ)๐๐ฅwhich be found at right
space the following ;
๐
๐๐ฅ
โซ ๐(๐ฅ)๐๐ฅ ยฑ โซ ๐(๐ฅ)๐๐ฅ =
๐
๐๐ฅ
[โซ ๐(๐ฅ)๐๐ฅ] ยฑ [โซ ๐(๐ฅ)๐๐ฅ] = ๐(๐ฅ) ยฑ ๐(๐ฅ)
๐
๐๐ฅ
โซ ๐(๐ฅ)๐๐ฅ ยฑ โซ ๐(๐ฅ)๐๐ฅ = ๐(๐ฅ) ยฑ ๐(๐ฅ)
So,
โซ (๐(๐ฅ) ยฑ ๐(๐ฅ))๐๐ฅ = โซ ๐(๐ฅ)๐๐ฅ ยฑ โซ ๐(๐ฅ)๐๐ฅ
1. Find integral from โซ (3๐ฅ2
โ 3๐ฅ + 7)๐๐ฅ!
Answers:
โซ (3๐ฅ2
โ 3๐ฅ + 7)๐๐ฅ = 3โซ ๐ฅ2
๐๐ฅ โ 3โซ ๐ฅ ๐๐ฅ + โซ 7 ๐๐ฅ theorema 2,3 and 4
=
3
2+1
๐ฅ2+1
โ
3
1+1
๐ฅ1+1
+ 7๐ฅ + ๐ theorem 1
5. = ๐ฅ3
โ
3
2
๐ฅ2
+ 7๐ฅ + ๐
So, โซ (3๐ฅ2
โ 3๐ฅ + 7)๐๐ฅ = ๐ฅ3
โ
3
2
๐ฅ2
+ 7๐ฅ + ๐
Prove Theorem 6
In the class XI, you have know derivative of two times product of functions ๐(๐ฅ) =
๐ข(๐ฅ). ๐ฃ(๐ฅ) is
๐
๐๐ฅ
[๐ข(๐ฅ). ๐ฃ(๐ฅ)] = ๐ข(๐ฅ). ๐ฃโฒ(๐ฅ) + ๐ฃ(๐ฅ). ๐ฃโฒ
(๐ฅ)
It will prove that partial integral rule with formula them. Method them is with differential
two equation it the following :
โซ
๐
๐๐ฅ
[๐ข(๐ฅ). ๐ฃ(๐ฅ)] = โซ ๐ข(๐ฅ). ๐ฃโฒ(๐ฅ) + โซ ๐ฃ(๐ฅ). ๐ฃโฒ
(๐ฅ)๐๐ฅ
๐ข(๐ฅ). ๐ฃ(๐ฅ) = โซ ๐ข(๐ฅ). ๐ฃโฒ(๐ฅ) + โซ ๐ฃ(๐ฅ). ๐ฃโฒ
(๐ฅ)dx
โซ ๐ข(๐ฅ). ๐ฃโฒ(๐ฅ) = ๐ข(๐ฅ). ๐ฃ(๐ฅ) โ โซ ๐ฃ(๐ฅ). ๐ฃโฒ
(๐ฅ)dx
Because vโ(x) dx= dv and uโ(x)dx=du
So,the equation can be written โซ ๐ข ๐๐ฃ = ๐ข๐ฃ โ โซ๐ฃ ๐๐ข
B.1 Substitution Integral Rule
Substitution Integral Rule is like which be written at Theorem 5. This rule was used
for to solve the problem in integration which not can to solve with base formulas what
already learn. For remainder it, example the following it
Example ;
1. Find the integral from
Answers;
a. Supposing that: u=9-x2
then du =-2x dx
6. So,
b. Supposing that u= โ ๐ฅ =๐ฅ
1
2
with the result that
c. Supposing that u= 1- 2x2
and du = -4x dx
dx =
๐๐ข
โ4๐ฅ
so the integral can be written the following
Substitution u= 1- 2x2
to equation 12u-3
+ c
7. So,
Prove theorem 7
In the class XI, you have learn derivative trigonometric function, is
๐
๐๐ฅ
(sin ๐ฅ) = cos ๐ฅ
๐
๐๐ฅ
(cos ๐ฅ) = โsin ๐ฅ ,and
๐
๐๐ฅ
(tan ๐ฅ) = ๐ ๐๐2
๐ฅ
The following this we can prove trigonometric integral rule to use formulas. This method is
with integration two space the following;
From
๐
๐๐ฅ
(sin ๐ฅ) = cos ๐ฅ be foundโซ ๐๐๐ ๐๐ฅ = sin ๐ฅ + ๐
From
๐
๐๐ฅ
(cos ๐ฅ) = โsin ๐ฅ ๐๐ ๐๐๐ข๐๐ โซ ๐ ๐๐ ๐๐ฅ = โcos ๐ฅ + ๐
From
๐
๐๐ฅ
(tan ๐ฅ) = ๐ ๐๐2
๐ฅ be found โซ ๐ ๐๐2
๐๐ฅ = tan ๐ฅ + ๐
B.2 space integral with โ๐ ๐ โ ๐ ๐, โ๐ ๐ + ๐ ๐ and โ๐ ๐ + ๐ ๐
Integration spaces โ๐2 โ ๐ฅ2, โ๐2 + ๐ฅ2 and โ๐ฅ2 + ๐2 can be work with substitution with x =
a sin t, x= a tan t, x = a sec t. So can be found spaces the following it ;
8. Right angle for integral trigonometric substitution;
(๐)โ๐2 โ ๐ฅ2 = ๐ cos ๐ฅ, (๐๐)โ๐2 + ๐ฅ2 = ๐ sec ๐ก, (๐๐๐)โ๐ฅ2 โ ๐2 =a tan x
1. Find each integral the following it:
Answers:
For to work this integral, you must change sin(3x+1)cos(3x+1) in the double angle
trigonometric formulas