THP-FTP-UB

Integration or antidifferentiation is thereverse process of differentiation.

The symbol 𝑓 𝑥 𝑑𝑥 denote the integral of

𝑓 𝑥 with respect to the variable 𝑥.

For example𝑑

𝑑𝑥𝑥4 = 4𝑥3, so the integral of

4𝑥3 with respect to 𝑥 is written by:

4𝑥3𝑑𝑥 = 𝑥4

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 symbol𝐶 to the result of integration:

4𝑥3𝑑𝑥 = 𝑥4 + 𝐶

𝑪 is called constant integration and must always beincluded.

Polynomial expression are integrated term byterm with the individual constant ofintegration consolidated into one symbol 𝐶 tofor whole expression.

Example

Integration of Functions of a Linier Function of 𝒙

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 3𝑥2+18𝑥+3

3𝑥2+5𝑥−2𝑑𝑥 by partial fraction

First we divide 3𝑥2 + 18𝑥 + 3 by 3𝑥2 + 5𝑥 − 2, so weget

Then, we solve 1 +13𝑥+5

3𝑥2+5𝑥−2𝑑𝑥 = 1𝑑𝑥 +

13𝑥+5

3𝑥2+5𝑥−2𝑑𝑥. To solve the form

13𝑥+5

3𝑥2+5𝑥−2𝑑𝑥 we just

can use the rule like previous example.

Find

Find

(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

𝑐𝑜𝑠2𝑥= 𝑠𝑒𝑐2𝑥, 𝑢 = 𝑥2,

𝑑𝑢

𝑑𝑥= 2𝑥, 𝑠𝑜

Evaluate

© (d)

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:

𝑠𝑖𝑛2𝑥 =1

21 − 𝑐𝑜𝑠2𝑥

𝑐𝑜𝑠2𝑥 =1

21 + 𝑐𝑜𝑠2𝑥

𝑠𝑖𝑛𝑥. 𝑐𝑜𝑠𝑥 =1

2𝑠𝑖𝑛2𝑥

For example: 2 1

sin 1 cos22

1 1cos2

2 2

sin 2

2 4

xdx x dx

dx xdx

x xC

Example Then we make substitution

if 𝑓integrable on 𝑎, 𝑏 , moreover 𝑎𝑏𝑓 𝑥 𝑑𝑥 , called

the definit integral of 𝑓 from 𝑎 to 𝑏.

Then 𝑎𝑏𝑓 𝑥 𝑑𝑥 = 𝐹 𝑏 − 𝐹 𝑎

which is 𝐹 be any antiderivative of 𝑓 on 𝑎, 𝑏

For example

−1

2

2𝑥 + 3𝑑𝑥 = 𝑥2 + 3𝑥 −12

= 22 + 3.2 − −1 2 + 3.−1 = 10 − −2 = 12

The techniques integration of definite integrals aresame with indefinite integral.

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