integration

1. University of Duhok
College of Basic Education
Dept.: Mathematic
((INTEGRATION))
-----------------------------------------------------------------------------------------------
Prepared by:
Parti Kazim Salih
1

2. Integration (from the Latin integer, meaning whole or entire) generally means combining parts
so that they work together or form a whole. In information technology, there are several
common usages:
………………………….
1) Integration during product development is a process in which separately produced
components or subsystems are combined and problems in their interactions are
addressed.
………………………..
2) Integration is an activity by companies that specialize in bringing different manufacturers'
products together into a smoothly working system.
2

3. Gottfried Wilhelm Leibniz
The modern notation for the indefinite integral was introduced by Gottfried Leibniz in
1675(Burton 1988,p.359;Liebinz 1899,p.154).He adapted the integral symbol .∫ , from
the letter (longs) standing for summa (written as summa ; Latin for “ sum ” or “total”.
The modern notation for the definite integral , with limits above and below the
integral sign , was first used by Joseph Fourier in Memoires of the French Academy
around
1819-20, reprinted in his book of 1822 ( Cajori1929,pp.249-250;Fourier 1822.§ 231
3

4. Newton and Leibniz
The major advance in integration came in the 17th century with the independent discovery of
the fundamental theorem of calculus by newton and Leibniz . The theorem demonstrates a
connection between integration and
Differentiation .this connection , combined with the comparative ease of differentiation , can be
exploited to calculate integrals .in particular , the fundamental theorem of calculus allows one
to solve a much broader class of problems .Equal in importance is the comprehensive
mathematical framework that both Newton and Leibniz developed . Given the name
infinitesimal calculus , it allowed for precise analysis of functions within continuous domain .
This framework eventually became modern calculus, whose notation for integrals is drawn
directly from the work of Leibniz.
4

5. Formula: 𝑥 𝑛
𝑑𝑥=
𝑥 𝑛+1
𝑛+1
+c ,
Proof: d(
𝑥 𝑛+1
𝑛+1
+c) =
𝑛+1 𝑥 𝑛+1−1
𝑛+1
𝑑𝑥
=
𝑛+1 𝑥 𝑛
(𝑛+1)
𝑑𝑥
=𝑥 𝑛
𝑑𝑥
Indefinite Integral
1) ∫adx=ax+c
Ex): ∫5dx=5x+c
…………………………….
2) ∫xⁿdx=
𝑥 𝑛+1
𝑛+1
+c ,n≠-1
Ex): ∫𝑥2
dx=
𝑥3
3
+c
……………………………..
3)∫(u±v±w)dx=∫udx±∫vdx±∫wdx
Ex):-∫(𝑥2+2x-4)dx
∫𝑥2
dx+∫2xdx-∫4dx
=
𝑥3
3
+
2𝑥2
2
− 4x + c ,,,,,, =
𝑥3
3
+𝑥2-4x+c 5

6. 4):
𝑢′
𝑢
dx=ln 𝑢 +c
Ex):-
1
𝑥
=ln 𝑥 +c
Ex):-
2𝑥
𝑥2=ln 𝑥2 +c
…………………………………………………………
5): 𝑒 𝑎𝑥
dx=
1
𝑎
𝑒 𝑎𝑥
+c
Ex):- 𝑒−2𝑥
dx=
1
−2
𝑒−2𝑥
+c
Ex):-
1
𝑥
dx
1
𝑥
1
2
𝑑𝑥 … … … … … . . 𝑥−
1
2 𝑑𝑥
=
𝑥−
1
2
+1
−
1
2 + 1
+ 𝑐
= 2𝑥
1
2 + 𝑐 … … … … … 2 𝑥 + 𝑐
6

7. 6) …. Integration of trigonometric function
a) 𝑠𝑖𝑛𝑥𝑑𝑥=-cosx +c , sin 𝑎𝑥dx=
−1
𝑎
cos 𝑎𝑥+c
b) 𝑐𝑜𝑠𝑥𝑑𝑥=sinx+c , cos 𝑎𝑥dx=
1
𝑎
sin 𝑎𝑥+c
c) 𝑠𝑒𝑐𝑎𝑥 . 𝑡𝑎𝑛𝑎𝑥 𝑑𝑥 =
1
𝑎
𝑠𝑒𝑐𝑎𝑥 + c
d) 𝑐𝑠𝑐𝑎𝑥. 𝑐𝑜𝑡𝑎𝑥 𝑑𝑥 = −
1
𝑎
𝑐𝑠𝑐𝑎𝑥 + 𝑐
e) 𝑠𝑒𝑐2axdx =
1
𝑎
𝑡𝑎𝑛𝑎𝑥 + 𝑐
f) 𝑐𝑠𝑐2
𝑎𝑥 𝑑𝑥 = −
1
𝑎
cot 𝑎𝑥+c…..
7

8. Ex):- (2cosx-5x)dx
2cos 𝑥dx- 5xdx
2 cos 𝑥dx-5 xdx
=2sin 𝑥-5
2
𝑥2 + 𝑐
……………………………………..
Ex):- 2 sin 𝑥𝑑𝑥
=2 sin 𝑥dx
=2*(-cos 𝑥+c
=-2cos 𝑥 + 𝑐
8

9. Ex) sin 3 − 5𝑥 𝑑𝑥 = −
1
5
(−cos 3 − 5x + c
=
1
5
cos 3 − 5x + c
Ex) cos 1 − 2𝑥 𝑑𝑥 = −
1
2
sin 1 − 2𝑥 + 𝑐
Ex)
1
𝑐𝑜𝑠2𝑥.𝑐𝑜𝑡2𝑥
𝑑𝑥 = 𝑠𝑒𝑐2𝑥. 𝑡𝑎𝑛2𝑥 𝑑𝑥
=
1
2
𝑠𝑒𝑐2𝑥 + 𝑐
Ex)
1
𝑐𝑜𝑠22𝑥
dx= 𝑠𝑒𝑐2
2𝑥 =
1
2
𝑡𝑎𝑛2𝑥 + 𝑐
9

10. Ex):- 2sin(3𝑥) 𝑑𝑥
2 sin(3𝑥) 𝑑𝑥
= 2 ∗ (−
1
3
cos(3𝑥)
= −
2
3
cos 3𝑥 + 𝑐
…………………………………………………………………
Ex):- 4 cos 2𝑥 𝑑𝑥
4 cos 2𝑥 𝑑𝑥
=
4
2
sin 2𝑥 … … . = 2 sin 2𝑥 + 𝑐
…………………………………………………………………
Ex):- (𝑡𝑎𝑛2
𝑥 + 1)dx
(
𝑠𝑖𝑛2 𝑥
𝑐𝑜𝑠2 𝑥
+1)dx *note(tan 𝑥 =
𝑠𝑖𝑛𝑥
cos 𝑥
)
𝑠𝑖𝑛2 𝑥+𝑐𝑜𝑠2 𝑥
𝑐𝑜𝑠2 𝑥
𝑑𝑥 note(𝑠𝑖𝑛2
𝑥 + 𝑐𝑜𝑠2
𝑥 = 1)
1
𝑐𝑜𝑠2 𝑥
=tanx+c 10

11. Ex):-
(−
9
𝑥4) =
3
𝑥3+c
𝑓 𝑥 =
3
𝑥3
𝑓 𝑥 = 3𝑥−3
𝑓′
𝑥 = −9𝑥−4
=
−9
𝑥4
11

12. 7): Definite Integral
𝑎
𝑏
𝑓 𝑥 𝑑𝑥 = F b − F a
………………………………………………………………………………………
𝑎
𝑏
𝑘 𝑓(𝑥)dx=k 𝑎
𝑏
𝑓(𝑥)dx
………………………………………………………………………………………
Ex):- 0
𝜋
(1 + 𝑠𝑖𝑛𝑥)𝑑𝑥
= 𝑥 − 𝑐𝑜𝑠𝑥 0
𝜋
= 𝜋 − 𝑐𝑜𝑠𝜋 − (0 − cos 0 )
= 𝜋+1-(-1)……… 𝜋+2
= 𝜋+1+1= 𝜋+2
12

13. Ex):- 2
4
𝑥2 − 3x − 2)dx
2
4
𝑥2
dx − 3
2
4
3𝑥𝑑𝑥 −
2
4
2𝑑𝑥
=
𝑥3
3 2
4
− 3
𝑥2
2 2
4
− 2 𝑥
=
43
3
−
23
3
− 3
4
2
2
−
2
2
2
− 2 4 − 2 =
−4
2
…………………………………………
Ex):- 0
𝜋
4
1−𝑠𝑖𝑛2 𝑥
𝑐𝑜𝑠2 𝑥
0
𝜋
4
𝑐𝑜𝑠2 𝑥
𝑐𝑜𝑠2 𝑥
**note{1−𝑠𝑖𝑛2
x}=𝑐𝑜𝑠2
𝑥**
0
𝜋
4
1𝑑𝑥 … . . . . . . … . . = 𝑥 0
𝜋
4
=
𝜋
4
−0=
𝜋
4
13

14. Ex):- Area: x=𝜋 , x=0 f(x)=sinx
A= 0
𝜋
𝑠𝑖𝑛𝑥 𝑑𝑥
= −𝑐𝑜𝑠𝑥 0
𝜋
=-cos𝜋-(-cos(0))
=-(-1)-(-1)
=1+1=2
……………………………………………………………………
Ex) 0
2𝜋
1 − 𝑠𝑖𝑛𝑥 𝑑𝑥
= x−(−cos x)
2𝜋
0
= (2𝜋+cos2𝜋) - ( 0 + cos(0))
= (2𝜋+1)-(1)
= 2𝜋 + 1 - 1 = 2𝜋
14

15. 𝑀𝑒𝑎𝑛 𝑣𝑎𝑙𝑢𝑒
𝑎, 𝑏 ,𝑓 𝑥 =
1
𝑎−𝑏 𝑎
𝑏
𝑓 𝑥 𝑑𝑥
Ex):-𝑓 𝑥 = (3𝑥2−2𝑥) 1,4
1
𝑎−𝑏 𝑎
𝑏
𝑓 𝑥 𝑑𝑥 ,,
1
1−4 1
4
(3𝑥2−2𝑥)𝑑𝑥
−
1
3
𝑥3
− 𝑥2 4
1
= −
1
3
64 − 16 − 1 − 1 = −16
15

16. Integration by substitution
𝑓 𝑥 𝑑𝑥 ………. 𝑢𝑑𝑢
Ex):-Find 9 𝑥2 + 3𝑥 + 5
8
2𝑥 + 3 𝑑𝑥
9𝑢8
𝑑𝑢 = 9
𝑢8+1
8 + 1
=9
𝑢9
9
= 𝑢9 + 𝑐
=(𝑥2
+ 3𝑥 + 5)9
+𝑐
……………………………………………………
Ex):-
(𝑥 + 4)5
𝑑𝑥
Let: u=(x+4)……….du=𝑢′
……du=dx
(𝑥 + 4)5
dx = 𝑢5
𝑑𝑢
=
(𝑢)6
6
+c
=
(𝑋+4)6
6
+c 16

17. Integration by part
∫udv=uv-∫vdu
Ex):- ∫xsin 𝑥 𝑑𝑥
u=x du=dx
dv=sinx v=-cosx
=x-cosx+∫cos 𝑥dx
=-xcos 𝑥 + 𝑠𝑖𝑛𝑥 + 𝑐
17

18. Ex):- 𝑥2
𝑒 𝑥
𝑑𝑥
u=𝑥2
,, du=2xdx ,, dv=𝑒 𝑥
, , v=𝑒 𝑥
𝑥2 𝑒 𝑥 𝑑𝑥 = 𝑥2 𝑒 𝑥 − 2𝑥 𝑒 𝑥 𝑑𝑥
= 𝑥2
𝑒 𝑥
− (2𝑥 𝑒 𝑒
− 2𝑒 𝑥
)+c
=𝑥2 𝑒 𝑥 − 2𝑥𝑒 𝑥 + 2𝑒 𝑥 + 𝑐
u=2x ,, du=2dx ,, dv=𝑒 𝑥
𝑑𝑥 ,, v=𝑒 𝑥
2𝑥 𝑒 𝑥 𝑑𝑥 = 2𝑥 𝑒 𝑥 − 2 𝑒 𝑥 𝑑𝑥
=2𝑥𝑒 𝑥
− 2 𝑒 𝑥
+c
18

19. Ex):- 0
𝜋
2 𝑥𝑐𝑜𝑠𝑥 dx
Let: u=x , du=dx , dv=cosx ,v=sinx
𝑢𝑑𝑣 = 𝑢𝑣 − 𝑣𝑑𝑢
=xsinx- 𝑠𝑖𝑛𝑥 𝑑𝑥
=xsinx-(-cosx)
=xsinx+cosx
0
𝜋
2
𝑥𝑐𝑜𝑠𝑥 𝑑𝑥 = 𝑥𝑠𝑖𝑛𝑥 + 𝑐𝑜𝑠𝑥
𝜋
2
0
(
𝜋
2
𝑠𝑖𝑛
𝜋
2
+cos
𝜋
2
) −(0*sin(0)+cos(0))
=
𝜋
2
(1)+0-(0+1) =
𝜋
2
-1
19

20. ex):- 𝑥3
𝑠𝑖𝑛𝑥 𝑑𝑥
u=𝑥3
, du=3𝑥2
, dv=𝑠𝑖𝑛𝑥 𝑑𝑥 , 𝑣 = −𝑐𝑜𝑠𝑥
𝑢𝑑𝑣 = 𝑢𝑣 − 𝑣𝑑𝑢
=𝑥3 −𝑐𝑜𝑠𝑥 − −𝑐𝑜𝑠𝑥 3𝑥2 𝑑𝑥
−𝑥3
𝑐𝑜𝑠𝑥 + 3𝑥2
𝑐𝑜𝑠𝑥𝑑 + 𝑐
u=3𝑥2 , du=6x , dv=cosx , v=sinx
−𝑥3
𝑐𝑜𝑠𝑥 + 3𝑥2
𝑠𝑖𝑛𝑥 − 𝑠𝑖𝑛𝑥 6𝑥 𝑑𝑥 + 𝑐
−𝑥3
𝑐𝑜𝑠𝑥 + 3𝑥2
𝑠𝑖𝑛𝑥 − 6𝑥𝑠𝑖𝑛𝑥 𝑑𝑥 + 𝑐
𝑢 = 6𝑥 , 𝑑𝑢 = 6𝑑𝑥 , 𝑑𝑣 = 𝑠𝑖𝑛𝑥 , 𝑣 = −𝑐𝑜𝑠𝑥
−𝑥3 𝑐𝑜𝑠𝑥 + 3𝑥2 𝑠𝑖𝑛𝑥 − 6x −cosx − −𝑐𝑜𝑠𝑥 6𝑑𝑥 + 𝑐
−𝑥3 𝑐𝑜𝑠𝑥 + 3𝑥2 𝑠𝑖𝑛𝑥 + 6𝑥 𝑐𝑜𝑠𝑥 − 6𝑠𝑖𝑛𝑥 + 𝑐
20

21. ex):- 𝑥2 𝑒−𝑥
𝑢 = 𝑥2 … … 𝑑𝑢 = 2𝑥𝑑𝑥 … 𝑣 = 𝑒−𝑥 … 𝑑𝑣 = −𝑒−𝑥
𝑢𝑑𝑣 = 𝑢𝑣 − 𝑣𝑑𝑢
=−𝑥2
𝑒−𝑥
+ 2 𝑥𝑒−𝑥
𝑑𝑥
𝑢 = 𝑥 … 𝑑𝑢 = 𝑑𝑥 … . 𝑑𝑣 = 𝑒−𝑥
… . . 𝑣 = −𝑒−𝑥
𝑥2 𝑒−𝑥 𝑑𝑥 = −𝑥2 𝑒−𝑥 + 2 𝑥 𝑒−𝑥 𝑑𝑥
=−𝑥2
𝑒−𝑥
+ 2(−𝑥 𝑒−𝑥
+ 𝑒−𝑥
𝑑𝑥)
=−𝑥2
𝑒−𝑥
− 2𝑥 𝑒−𝑥
− 2 𝑒−𝑥
+ 𝑐
21