Basic mathematics integration
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Transcript of Basic mathematics integration
Integration or antidifferentiation is thereverse process of differentiation.
The symbol π π₯ ππ₯ denote the integral of
π π₯ with respect to the variable π₯.
For exampleπ
ππ₯π₯4 = 4π₯3, so the integral of
4π₯3 with respect to π₯ is written by:
4π₯3ππ₯ = π₯4
See that
Because any constant term in the originalexpression becomes zero in the derivative. Wetherefore acknowledge the presence of suchconstant term of some value by adding a symbolπΆ to the result of integration:
4π₯3ππ₯ = π₯4 + πΆ
πͺ is called constant integration and must always beincluded.
Polynomial expression are integrated term byterm with the individual constant ofintegration consolidated into one symbol πΆ tofor whole expression.
Example
Integration of Functions of a Linier Function of π
If:
then:
For example:
( ) ( ) f x dx F x C
( )( )
F ax bf ax b dx C
a
7 76 6 (5 4)
so that (5 4)7 7 5
x x
x dx C x dx C
If the integrand is an algebraic fraction thatcan be separated into its partial fractionsthen each individual partial fraction can beintegrated separately.
2
1 3 2
3 2 2 1
3 2
2 1
3ln | 2 | 2ln | 1|
xdx dx
x x x x
dx dxx x
x x C
If the numerator is not of lower degree than thedenominator, the first step is to divide out.
For example
Determine 3π₯2+18π₯+3
3π₯2+5π₯β2ππ₯ by partial fraction
First we divide 3π₯2 + 18π₯ + 3 by 3π₯2 + 5π₯ β 2, so weget
Then, we solve 1 +13π₯+5
3π₯2+5π₯β2ππ₯ = 1ππ₯ +
13π₯+5
3π₯2+5π₯β2ππ₯. To solve the form
13π₯+5
3π₯2+5π₯β2ππ₯ we just
can use the rule like previous example.
(i)
For example
(ii)
For example
( ) 1( ) ln ( )
( ) ( )
f xdx df x f x C
f x f x
2
2
2 2
2 3 ( 3 5)ln 3 5
3 5 3 5
x d x x
dx x x Cx x x x
(iii)
Example
Since 1
πππ 2π₯= π ππ2π₯, π’ = π₯2,
ππ’
ππ₯= 2π₯, π π
The part formula is
For example
( ) ( ) ( ) ( ) ( ) ( ) u x dv x u x v x v x du x
( ) ( )
( ) ( ) ( ) ( ) where ( ) so ( )
( ) so ( )
.
x
x x
x x
x x
xe dx u x dv x
u x v x v x du x u x x du x dx
dv x e dx v x e
x e e dx
xe e C
Many integrals with trigonometric integrands canbe evaluated after applying trigonometricidentities.
Trigonometric identities such as:
π ππ2π₯ =1
21 β πππ 2π₯
πππ 2π₯ =1
21 + πππ 2π₯
π πππ₯. πππ π₯ =1
2π ππ2π₯
For example: 2 1
sin 1 cos22
1 1cos2
2 2
sin 2
2 4
xdx x dx
dx xdx
x xC
if πintegrable on π, π , moreover πππ π₯ ππ₯ , called
the definit integral of π from π to π.
Then πππ π₯ ππ₯ = πΉ π β πΉ π
which is πΉ be any antiderivative of π on π, π
For example
β1
2
2π₯ + 3ππ₯ = π₯2 + 3π₯ β12
= 22 + 3.2 β β1 2 + 3.β1 = 10 β β2 = 12
The techniques integration of definite integrals aresame with indefinite integral.