Fjfjdjejrkdidjxjsjrkkdjdkkjjkkfkckckdkükememememdjdjdjjdnrnene3irjrjrjüjejsjdndnenejejdjdjdjdjdjsjsjejrururirjrjrjddjdjjdjjrjrjrjrjriejrjrjridududjrjrjejejeididjdjdje**Introduction:** If you want to focus better and get more things done, there are two simple solutions to consider. First, turn off notifications on your phone, especially from news apps and emails. Second, try deleting Facebook from your smartphone – it can help you avoid wasting time. **Body:** 1. *Turn Off Notifications:* To concentrate on your tasks, start by stopping notifications from bothering you. Turn off those pop-up messages from news apps and emails. This way, you won't be interrupted while working or studying. 2. *Delete Facebook:* Consider removing Facebook from your phone. It might seem difficult at first, but it can help you stop spending too much time scrolling through posts. This small change can give you more free time. **Conclusion:** These ideas won't magically make you super productive, but they can make your life a bit simpler. By managing notifications and removing distractions like Facebook, you can focus on what's important in your life.**Introduction:** If you want to focus better and get more things done, there are two simple solutions to consider. First, turn off notifications on your phone, especially from news apps and emails. Second, try deleting Facebook from your smartphone – it can help you avoid wasting time. **Body:** 1. *Turn Off Notifications:* To concentrate on your tasks, start by stopping notifications from bothering you. Turn off those pop-up messages from news apps and emails. This way, you won't be interrupted while working or studying. 2. *Delete Facebook:* Consider removing Facebook from your phone. It might seem difficult at first, but it can help you stop spending too much time scrolling through posts. This small change can give you more free time. **Conclusion:** These ideas won't magically make you super productive, but they can make your life a bit simpler. By managing notifications and removing distractions like Facebook, you can focus on what's important in your life.**Introduction:** If you want to focus better and get more things done, there are two simple solutions to consider. First, turn off notifications on your phone, especially from news apps and emails. Second, try deleting Facebook from your smartphone – it can help you avoid wasting time. **Body:** 1. *Turn Off Notifications:* To concentrate on your tasks, start by stopping notifications from bothering you. Turn off those pop-up messages from news apps and emails. This way, you won't be interrupted while working or studying. 2. *Delete Facebook:* Consider removing Facebook from your phone. It might seem difficult at first, but it can help you stop spending too much time scrolling through posts. This small change can give you more free time. **Conclusion:** These ideas won't magically make you super productive, but they can make your life a bit

1. 15.Rasional, irrasional və
transendent funksiyalarin
inteqrallanması.

2. Misal 1:
əvəzləməsi aparaq:
      
 
 





x
x
x
x
x
x
x
e
d
e
e
dx
e
e
dx
e
e 2
2
2
2
2
2
2
4
2
4
4
1
2
1
4
16
x
e
t 2

      






  

















dt
t
dt
t
dt
t
t
dt
t
t
dx
e
e
x
x
4
1
16
1
4
1
16
1
4
1
4
1
16
1
4
4
1
2
1
16
4
2
C
e
e
C
t
t
C
t
t x
x













4
4
ln
16
1
4
4
ln
16
1
4
ln
16
1
4
ln
16
1
2
2

3. Misal2:
2𝑥4+3𝑥2+𝑥+1
𝑥 𝑥2+1
𝑑𝑥 =
2𝑥2 𝑥2+1 + 𝑥2+1 +𝑥
𝑥 𝑥2+1
𝑑𝑥 =
=
2𝑥2 𝑥2+1
𝑥 𝑥2+1
+
𝑥2+1
𝑥 𝑥2+1
+
𝑥
𝑥 𝑥2+1
𝑑𝑥 =
= 2𝑥 +
1
𝑥
+
1
𝑥2+1
𝑑𝑥 = 2 𝑥𝑑𝑥 +
1
𝑥
𝑑𝑥 +
1
𝑥2+1
𝑑𝑥 =
= 𝑥2 +𝑙𝑛 𝑥 + 𝑎𝑟𝑐𝑡𝑔𝑥 + 𝑐.
𝐌𝐢𝐬𝐚𝐥 𝟑:
𝑑𝑥
4𝑥2 + 25
=
1
4
𝑑𝑥
𝑥2 +
25
4
=
1
4
𝑑𝑥
𝑥2 +
5
2
2 =
1
4
∙
1
5
2
∙ 𝑎𝑟𝑐𝑡𝑔
𝑥
5
2
+ 𝑐
=
=
1
10
∙ 𝑎𝑟𝑐𝑡𝑔
2𝑥
5
+ 𝑐.

4. 𝐌𝐢𝐬𝐚𝐥 𝟒:
𝑥2
𝑥2 + 4
𝑑𝑥 = 1 −
4
𝑥2 + 4
𝑑𝑥 = 1𝑑𝑥 −
4
𝑥2 + 4
𝑑𝑥 =
= 𝑥 − 4 ∙
1
2
∙ 𝑎𝑟𝑐𝑡𝑔
𝑥
2
+ 𝑐 = 𝑥 − 2 ∙ 𝑎𝑟𝑐𝑡𝑔
𝑥
2
+ 𝑐.
Misal 5:
𝑥5+2𝑥3+4𝑥+4
𝑥4+2𝑥3+2𝑥2 𝑑𝑥
𝑥5+2𝑥3+4𝑥+4
𝑥4+2𝑥3+2𝑥2 – düzgün olmayan rasional kəsrdir. x5 + 2x3 + 4x + 4
çoxhədlisini x4 + 2x3 + 2x2 çoxhədlisinə bölək. Nəticədə alırıq:
x5 + 2x3 + 4x + 4 = (x – 2)(x4 + 2x3 + 2x2) + 4x3 + 4x2 + 4x + 4

5. Yəni
𝑥5 + 2𝑥3 + 4𝑥 + 4
𝑥4 + 2𝑥3 + 2𝑥2
𝑑𝑥 = 𝑥 − 2 +
4𝑥3 + 4𝑥2 + 4𝑥 + 4
𝑥4 + 2𝑥3 + 2𝑥2
𝑑𝑥 =
= 𝑥 − 2 𝑑𝑥 +
4𝑥3+4𝑥2+4𝑥+4
𝑥4+2𝑥3+2𝑥2 𝑑𝑥 (1)
4𝑥3 + 4𝑥2 + 4𝑥 + 4
𝑥4 + 2𝑥3 + 2𝑥2
=
4𝑥3 + 4𝑥2 + 4𝑥 + 4
𝑥2(𝑥2 + 2𝑥 + 2)
=
𝐴
𝑥
+
𝐵
𝑥2
+
𝐶𝑥 + 𝐷
𝑥2 + 2𝑥 + 2
=
=
𝐴𝑥 𝑥2 + 2𝑥 + 2 + 𝐵 𝑥2 + 2𝑥 + 2 + 𝑥(𝐶𝑥 + 𝐷)
𝑥2(𝑥2 + 2𝑥 + 2)
=
=
𝐴+𝐶 𝑥3+ 2𝐴+𝐵+𝐷 𝑥2+ 2𝐴+2𝐵 𝑥+2𝐵
𝑥2(𝑥2+2𝑥+2)

6. 𝐴 + 𝐶 = 4
2𝐴 + 𝐵 + 𝐷 = 4
2𝐴 + 2𝐵 = 4
2𝐵 = 4
 B = 2 , A = 0 , C = 4 , D = 2
4𝑥3+4𝑥2+4𝑥+4
𝑥4+2𝑥3+2𝑥2 =
2
𝑥2 +
4𝑥+2
𝑥2+2𝑥+2
(2)
(2)-ni (1)-də yazaq:
𝑥5 + 2𝑥3 + 4𝑥 + 4
𝑥4 + 2𝑥3 + 2𝑥2
𝑑𝑥 = 𝑥 − 2 𝑑𝑥 +
2
𝑥2
+
4𝑥 + 2
𝑥2 + 2𝑥 + 2
𝑑𝑥 =
=
𝑥2
2
− 2𝑥 −
2
𝑥
+
4𝑥+2
𝑥2+2𝑥+2
𝑑𝑥 (3)
4𝑥 + 2
𝑥2 + 2𝑥 + 2
𝑑𝑥 =
4𝑥 + 2
(𝑥 + 1)2+1
𝑑𝑥

7. t = x + 1 əvəzləməsi aparaq:
t = x + 1  x = t – 1  dx = dt
4𝑥+2
(𝑥+1)2+1
𝑑𝑥 = 4
𝑡𝑑𝑡
𝑡2+1
− 2
𝑑𝑡
𝑡2+1
= 2
1
𝑡2+1
𝑑 𝑡2 + 1 − 2𝑎𝑟𝑐𝑡𝑔 𝑡 +
𝐶 =
= 2 ln(t2 + 1) – 2 arctg t + C = 2 ln(x2 + 2x + 2) – 2 arctg(x + 1) + C
(4)
(4)-ü (3)-də nəzərə alaq:
𝑥5
+ 2𝑥3
+ 4𝑥 + 4
𝑥4 + 2𝑥3 + 2𝑥2
𝑑𝑥
=
𝑥2
2
− 2𝑥 −
2
𝑥
+ 2𝑙𝑛 𝑥2 + 2𝑥 + 2 − 2𝑎𝑟𝑐𝑡𝑔 𝑥 + 1 + 𝐶

8. Misal 6:
𝑥2−2𝑥+2
𝑥3+2𝑥2−8𝑥
𝑑𝑥
𝑥2 − 2𝑥 + 2
𝑥3 + 2𝑥2 − 8𝑥
=
𝑥2 − 2𝑥 + 2
𝑥(𝑥 + 4)(𝑥 − 2)
=
𝐴
𝑥
+
𝐵
𝑥 − 2
+
𝐶
𝑥 + 4
=
=
𝐴 𝑥 − 2 𝑥 + 4 + 𝐵𝑥 𝑥 + 4 + 𝐶𝑥 𝑥 − 2
𝑥3 + 2𝑥2 − 8𝑥
=
=
𝐴+𝐵+𝐶 𝑥2+ 2𝐴+4𝐵−2𝐶 𝑥−8𝐴
𝑥3+2𝑥2−8𝑥


𝐴 + 𝐵 + 𝐶 = 1
2𝐴 + 4𝐵 − 2𝐶 = −2
−8𝐴 = 2

𝐴 = −
1
4
𝐶 =
13
12
𝐵 =
1
6

9. 𝑥2 − 2𝑥 + 2
𝑥3 + 2𝑥2 − 8𝑥
𝑑𝑥 = −
1
4
𝑥 +
1
6
∙
1
𝑥 − 2
+
13
12
∙
1
𝑥 + 4
𝑑𝑥 =
= −
1
4
1
𝑥
𝑑𝑥 +
1
6
1
𝑥 − 2
𝑑𝑥 +
13
12
1
𝑥 + 4
𝑑𝑥 =
= −
1
4
ln 𝑥 +
1
6
ln 𝑥 − 2 +
13
12
ln 𝑥 + 4 + 𝐶.
Misal 7: 𝐽3 =
𝑑𝑡
𝑡2+1 3
Qeyd edək ki, 𝐽𝑘 =
𝑑𝑡
𝑡2+𝑎2 𝑘 =
1
𝑎2
2𝑘−3
2𝑘−2
𝐽𝑘−1 +
𝑡
2 𝑘−1 𝑡2+𝑎2 𝑘−1
Baxılan misalda 𝑎 = 1, 𝑘 = 3 – dür.

10. 𝐽1 =
𝑑𝑡
𝑡2+1
= 𝑎𝑟𝑐𝑡𝑔𝑡 + 𝑐
𝐽2 =
𝑑𝑡
𝑡2+1 2 =
1
2
𝐽1 +
𝑡
2 𝑡2+1
=
1
2
𝑎𝑟𝑐𝑡𝑔𝑡 +
𝑡
2 𝑡2+1
+ 𝑐
𝐽3 =
3
4
𝐽2 +
𝑡
4 𝑡2+1 2 =
3
4
1
2
𝑎𝑟𝑐𝑡𝑔𝑡 +
𝑡
2 𝑡2+1
+
𝑡
4 𝑡2+1 2 + 𝑐
Misal 8:
𝑑𝑥
3+𝑠𝑖𝑛𝑥+𝑐𝑜𝑠𝑥
𝑡 = 𝑡𝑔
𝑥
2
əvəzləməsi aparaq:
𝑥
2
= 𝑎𝑟𝑐𝑡𝑔𝑡 ⇒ 𝑥 = 2𝑎𝑟𝑐𝑡𝑔𝑡 ⇒ 𝑑𝑥 =
2
𝑡2 + 1
𝑑𝑡
Qeyd edək ki,

11. 𝑠𝑖𝑛𝑥 =
2𝑡𝑔
𝑥
2
1 + 𝑡𝑔2 𝑥
2
=
2𝑡
1 + 𝑡2
𝑐𝑜𝑠𝑥 =
1 − 𝑡𝑔2 𝑥
2
1 + 𝑡𝑔2 𝑥
2
=
1 − 𝑡2
1 + 𝑡2
𝑑𝑥
3 + 𝑠𝑖𝑛𝑥 + 𝑐𝑜𝑠𝑥
=
2𝑑𝑡
1 + 𝑡2 3 +
2𝑡
1 + 𝑡2 +
1 − 𝑡2
1 + 𝑡2
=
𝑑𝑡
𝑡2 + 𝑡 + 2
=
=
𝑑 𝑡 +
1
2
𝑡 +
1
2
2
+
7
2
2 =
𝑑𝑦
𝑦2 +
7
2
2 =
1
7
2
∙ 𝑎𝑟𝑐𝑡𝑔
𝑦
7
2
+ 𝑐 =

12. =
2
7
𝑎𝑟𝑐𝑡𝑔
2𝑦
7
+ 𝑐 =
2
7
𝑎𝑟𝑐𝑡𝑔
2 𝑡 +
1
2
7
+ 𝑐
=
2
7
𝑎𝑟𝑐𝑡𝑔
2𝑡 + 1
7
+ 𝑐 =
=
2
7
𝑎𝑟𝑐𝑡𝑔
2𝑔
𝑥
2
+ 1
7

13. Misal 9: sin4 𝑥 cos5 𝑥𝑑𝑥
𝑡 = 𝑠𝑖𝑛𝑥 əvəzləməsi aparaq:
𝑥 = 𝑎𝑟𝑐𝑠𝑖𝑛𝑡 ⇒ 𝑑𝑥 =
1
1 − 𝑡2
𝑑𝑡 ⇒ 𝑐𝑜𝑠𝑥 = 1 − 𝑡2
sin4 𝑥 cos5 𝑥𝑑𝑥 = 𝑡4 1 − 𝑡2
5
∙
1
1 − 𝑡2
𝑑𝑡 = 𝑡4 1 − 𝑡2 2𝑑𝑡 =
= 𝑡4 − 2𝑡6 + 𝑡8 𝑑𝑡 =
1
5
𝑡5 −
2
7
𝑡7 +
1
9
𝑡9 + 𝑐 =
=
1
5
𝑠𝑖𝑛5𝑥 −
2
7
𝑠𝑖𝑛7𝑥 +
1
9
𝑠𝑖𝑛9𝑥 + 𝑐.

14. 𝐌𝐢𝐬𝐚𝐥𝟏𝟎: sin4 𝑥 cos2 𝑥𝑑𝑥 = 𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑥 2 sin2 𝑥𝑑𝑥 =
=
1
4
sin2 2𝑥 ∙
1 − 𝑐𝑜𝑠2𝑥
2
𝑑𝑥 =
1
8
sin2 2𝑥𝑑𝑥 −
−
1
8
sin2
2 𝑥 𝑐𝑜𝑠2 𝑥𝑑𝑥 =
1
8
1 − 𝑐𝑜𝑠4𝑥
2
𝑑𝑥 −
1
8
sin2
2𝑥𝑑 𝑠𝑖𝑛2𝑥 =
=
1
16
1 − 𝑐𝑜𝑠4𝑥 𝑑𝑥 −
1
16
sin22𝑥𝑑 𝑠𝑖𝑛2𝑥 =
1
16
1𝑑𝑥 −
1
16
𝑐𝑜𝑠4𝑥𝑑𝑥 −
−
1
16
sin2
2𝑥𝑑 𝑠𝑖𝑛2𝑥 =
1
16
𝑥 −
1
64
𝑠𝑖𝑛4𝑥 −
1
48
sin3
2𝑥 + 𝑐.

15. 𝐌𝐢𝐬𝐚𝐥𝟏𝟏:
𝑑𝑥
𝑐𝑜𝑠𝑥sin3𝑥
= 𝑐𝑜𝑠𝑥 −1
∙ 𝑠𝑖𝑛𝑥 −3
𝑑𝑥
𝑚 + 𝑛 = −4 olduğuna əsasən 𝑡 = 𝑡𝑔𝑥 əvəzləməsi aparaq:
𝑡 = 𝑡𝑔𝑥 ⇒ 𝑥 = 𝑎𝑟𝑐𝑡𝑔𝑡 ⇒ 𝑑𝑥 =
𝑑𝑡
1 + 𝑡2
𝑠𝑖𝑛𝑥 =
𝑡𝑔𝑥
1 + 𝑡𝑔2𝑥
=
𝑡
1 + 𝑡2
𝑐𝑜𝑠𝑥 =
1
1 + 𝑡𝑔2𝑥
=
1
1 + 𝑡2
Nəticədə alırıq:

16. 𝑑𝑥
𝑐𝑜𝑠𝑥sin3𝑥
=
𝑑𝑡
1 + 𝑡2
1
1 + 𝑡2
∙
𝑡3
1 + 𝑡2
3
=
1 + 𝑡2
𝑡3
𝑑𝑡 = 𝑡−3𝑑𝑡 +
𝑑𝑡
𝑡
=
= −
1
2𝑡2 + 𝑙𝑛 𝑡 + 𝑐 = −
1
2
𝑐𝑡𝑔2𝑥 + 𝑙𝑛 𝑡𝑔𝑥 + 𝑐.
Misal 12: 𝑠𝑖𝑛8𝑥𝑐𝑜𝑠2𝑥𝑑𝑥 =
1
2
𝑠𝑖𝑛10𝑥 + 𝑠𝑖𝑛6𝑥 𝑑𝑥 =
1
2
𝑠𝑖𝑛10𝑥𝑑𝑥 +
+
1
2
𝑠𝑖𝑛6𝑥𝑑𝑥 =
1
2
−
1
10
𝑐𝑜𝑠10𝑥 − −
1
6
𝑐𝑜𝑠6𝑥 +c

17. Misal 13:
 
  




 


 xdx
xdx
dx
x
x
xdx
x 6
sin
2
1
12
sin
2
1
6
sin
12
sin
2
1
9
cos
3
sin
C
x
x 


 6
cos
12
1
12
cos
24
1

18. Misal 14:
𝑑𝑥
𝑥2+𝑎2
𝑡 = 𝑥2 + 𝑎2 + 𝑥 əvəzləməsi aparaq:
𝑑𝑡 =
2𝑥
2 𝑥2 + 𝑎2
𝑑𝑥 + 𝑑𝑥 ⇒ 𝑑𝑡 =
𝑥2 + 𝑎2 + 𝑥
𝑥2 + 𝑎2
𝑑𝑥 ⇒ 𝑑𝑥
=
𝑥2 + 𝑎2
𝑥2 + 𝑎2 + 𝑥
𝑑𝑡
𝑑𝑥
𝑥2 + 𝑎2
=
1
𝑥2 + 𝑎2
𝑑𝑥 =
1
𝑥2 + 𝑎2
∙
𝑥2 + 𝑎2
𝑥2 + 𝑎2 + 𝑥
𝑑𝑡 =
=
1
𝑥2+𝑎2+𝑥
𝑑𝑡 =
1
𝑡
𝑑𝑡 = 𝑙𝑛 𝑡 + 𝑐 = 𝑙𝑛 𝑥2 + 𝑎2 + 𝑥 + 𝑐 .

19. Misal 15:
𝑑𝑥
4𝑥2+1
=
𝑑𝑥
4 𝑥2+
1
4
=
1
2
𝑑𝑥
𝑥2+
1
4
=
1
2
𝑙𝑛 𝑥 + 𝑥2 +
1
4
+ 𝑐
Misal 16:
4−𝑥2
𝑥2 𝑑𝑥
x=2sin t əvəzləməsi apraq:
𝑥 = 2 sin 𝑡 ⇒ 𝑑𝑥 = 2 cos 𝑡 𝑑𝑡 ⇒ 𝑡 = arcsin
𝑥
2

20. 4 − 𝑥2
𝑥2
𝑑𝑥 =
4 − 4𝑠𝑖𝑛 2𝑡
4𝑠𝑖𝑛2𝑡
∙ 2 cos 𝑡 𝑑𝑡 =
𝑐𝑜𝑠2
𝑡
𝑠𝑖𝑛2𝑡
𝑑𝑡 =
=
1
𝑠𝑖𝑛 2𝑡
− 1 𝑑𝑡 =
=
1
𝑠𝑖𝑛 2𝑡
𝑑𝑡 − 𝑑𝑡 = −𝑐𝑡𝑔 𝑡 − 𝑡 + 𝐶 == − arcsin
𝑥
2
− 𝑐𝑡𝑔 arcsin
𝑥
2
+
+𝐶 = − arcsin
𝑥
2
−
4 − 𝑥2
𝑥
+ 𝐶

21. 𝑐𝑡𝑔 𝑡 =
1 − 𝑠𝑖𝑛 2𝑡
sin 𝑡
=
1 −
𝑥
2
2
𝑥
2

22. Misal 17:
𝑥2+2𝑥−4
𝑥+1 3 𝑑𝑥
𝑥2 + 2𝑥 − 4 = 𝑥 + 1 2 − 5
t=x+1 əvəzləməsi aparaq:
𝑥 = 𝑡 − 1 ⇒ 𝑑𝑥 = 𝑑𝑡
𝑥2 + 2𝑥 − 4
𝑥 + 1 3
𝑑𝑥 =
𝑡2 − 5
𝑡3
𝑑𝑡
2-ci 𝑡 =
5
sin 𝑧
əvəzləməsi aparaq:
𝑑𝑡 =
− 5 cos 𝑧
𝑠𝑖𝑛2𝑧
𝑑𝑧
𝑧 = 𝑎𝑟𝑐𝑠𝑖𝑛
5
𝑡

23. 𝑥2 + 2𝑥 − 4
𝑥 + 1 3
𝑑𝑥 =
𝑥 + 1 2 − 5
𝑥 + 1 3
𝑑𝑥 =
𝑡2 − 5
𝑡3
𝑑𝑡 =
=
5
sin 𝑧
2
− 5
5
sin 𝑧
3 ∙
− 5 cos 𝑧
𝑠𝑖𝑛2𝑧
𝑑𝑧 =
= −
1
5
𝑐𝑜𝑠 2𝑧 𝑑𝑧 = −
1
2 5
1 + cos2 𝑧 𝑑𝑧 = −
5
10
𝑧 +
1
2
sin 2𝑧 + 𝐶 =
= −
5
10
arcsin
5
𝑡
+
1
2
𝑠𝑖𝑛 2 𝑎𝑟𝑐𝑠𝑖𝑛
5
𝑡
+ 𝐶=

24. =−
5
10
arcsin
5
𝑡
+ 𝑠𝑖𝑛 𝑎𝑟𝑐𝑠𝑖𝑛
5
𝑡
∙ 𝑐𝑜𝑠 𝑎𝑟𝑐𝑠𝑖𝑛
5
𝑡
+ 𝐶 =
= −
5
10
arcsin
5
𝑥 + 1
+
5
𝑥 + 1
∙
𝑥2 + 2𝑥 − 4
𝑥 + 1
+ 𝐶 =
= −
5
10
arcsin
5
𝑥 + 1
+
5𝑥2 + 10𝑥 − 20
𝑥 + 1 2
+ 𝐶

25. Misal 18:
𝑡𝑔 𝑥
1−𝑐𝑡𝑔2𝑥
𝑑𝑥
𝑡 = 𝑡𝑔 𝑥 əvəzləməsi aparaq:
𝑡 = 𝑡𝑔 𝑥 ⇒ 𝑑𝑡 =
1
𝑐𝑜𝑠2𝑥
𝑑𝑥 ⇒ 𝑑𝑥 = 𝑐𝑜𝑠2𝑥𝑑𝑡 ⇒ 𝑑𝑥 =
1
1 + 𝑡𝑔2𝑥
𝑑𝑡
𝑑𝑥 =
1
1 + 𝑡2
𝑑𝑡
𝑡𝑔 𝑥
1 − 𝑐𝑡𝑔2𝑥
𝑑𝑥 =
𝑡
1 −
1
𝑡2
∙
1
1 + 𝑡2
𝑑𝑡 =
𝑡3
𝑡4 − 1
𝑑𝑡 =
1
4
𝑑 𝑡4 − 1
𝑡4 − 1
=
=
1
4
𝑙𝑛 𝑡4 − 1 + 𝐶 =
1
4
𝑙𝑛 𝑡𝑔4𝑥 − 1 + 𝐶

26. Misal 19:
𝑑𝑥
4𝑥2+2𝑥+1
=
𝑑𝑥
2 𝑥2+
1
2
𝑥+4
=
1
2
𝑑𝑥
𝑥+
1
4
2
+
3
16
𝑡 = 𝑥 +
1
4
əvəzləməsi aparaq:
𝑑𝑥 = 𝑑𝑡
𝑑𝑥
4𝑥2 + 2𝑥 + 1
=
1
2
𝑑𝑥
𝑡2 +
3
16
=
1
2
𝑙𝑛 𝑡 + 𝑡2 +
3
16
+ 𝑐 =
=
1
2
𝑙𝑛 𝑥 +
1
4
+ 𝑥 +
1
4
2
+
3
16
+ 𝑐

27. Misal 20:
𝑥+4
6−2𝑥−𝑥2
𝑑𝑥 =
𝑥+4
7− 𝑥+1 2
𝑑𝑥
𝑡 = 𝑥 + 1 əvəzləməsi aparaq:
𝑥 = 𝑡 − 1 ⇒ 𝑑𝑥 = 𝑑𝑡
𝑥 + 4
6 − 2𝑥 − 𝑥2
𝑑𝑥 =
𝑡 + 3
7 − 𝑡2
𝑑𝑡 =
𝑡𝑑𝑡
7 − 𝑡2
+
𝑑𝑡
7 − 𝑡2
=
= −
1
2
7 − 𝑡2 −
1
2𝑑 7 − 𝑡2 + 3
𝑑𝑡
7
2
− 𝑡2
=
= − 7 − 𝑡2 + 3𝑎𝑟𝑐𝑠𝑖𝑛
𝑡
7
+ 𝑐 = 3𝑎𝑟𝑐𝑠𝑖𝑛
𝑥 + 1
7
− 6 − 2𝑥 − 𝑥2 + 𝑐

28. Misal 21:
𝑑𝑥
3
𝑥+2 2− 𝑥+2
=
𝑑𝑥
𝑥+2
2
3− 𝑥+2
1
2
2
3
𝑣ə
1
2
kəsrlərinin ən kiçik ortaq bölünəni 6 olduğuna əsasən
𝑥 + 2 = 𝑡6 əvəzləməsi aparaq:
𝑥 + 2 = 𝑡6
⇒ 𝑥 = 𝑡6
− 2 ⇒ 𝑑𝑥 = 6𝑡5
𝑑𝑡 ⇒ 𝑡 =
6
𝑥 + 2
Yəni,
𝑑𝑥
3
𝑥 + 2 2 − 𝑥 + 2
=
6𝑡5𝑑𝑡
𝑡4 − 𝑡3
= 6
𝑡2𝑑𝑡
𝑡 − 1
= 6
𝑡2 − 1 + 1
𝑡 − 1
𝑑𝑡 =
= 6
𝑡2 − 1
𝑡 − 1
+
1
𝑡 − 1
𝑑𝑡 = 6 𝑡 + 1 +
1
𝑡 − 1
𝑑𝑡 =
= 3𝑡2
+ 6𝑡 + 6𝑙𝑛 𝑡 − 1 + 𝑐 = 3
3
𝑥 + 2 + 6
6
𝑥 + 2 + 6𝑙𝑛
6
𝑥 + 2 − 1 + 𝑐.

29. Misal 22:
𝑑𝑥
𝑥4 1+𝑥2
= 𝑥−4 1 + 𝑥2 −
1
2𝑑𝑥
𝑚 = −4; 𝑛 = 2; 𝑝 = −
1
2
və
𝑚+1
𝑛
+ 𝑝 = −2-dir.
𝑥−2
+ 1 = 𝑡2
əvəzləməsi aparaq:
−2𝑥−3
𝑑𝑥 = 2𝑡 𝑑𝑡 ⇒ 𝑥−3
𝑑𝑥 = −𝑡𝑑𝑡
𝑑𝑥
𝑥4 1 + 𝑥2
= 𝑥−4 1 + 𝑥2 −
1
2𝑑𝑥 = 𝑥−4 𝑥2 𝑥−2 + 1
−
1
2𝑑𝑥 =
= 𝑥−2 𝑥−2 + 1 −
1
2𝑥−3𝑑𝑥 = − 𝑡2 − 1 𝑡−1𝑡𝑑𝑡 = 1 − 𝑡2 𝑑𝑡 =
= 𝑡 −
1
3
𝑡3 + 𝐶 = 𝑥−2 + 1 −
1
3
𝑥−2 + 1 3 + 𝐶 ==
𝑥2 + 1
𝑥
−
1 + 𝑥2 3
3𝑥3
+ 𝐶

30. Misal 23:
3 4
𝑥+1
𝑥
𝑑𝑥 =
𝑥
1
4+1
1
3
𝑥
1
2
𝑑𝑥 = 𝑥−
1
2 1 + 𝑥
1
4
1
3
𝑑𝑥
𝑚 = −
1
2
; 𝑛 =
1
4
; 𝑝 =
1
3
və
𝑚+1
𝑛
= 2-dir.
4
𝑥 + 1 = 𝑡3 əvəzləməsi aparaq:
𝑥 = (𝑡3
− 1)4
⇒ 𝑑𝑥 = 4(𝑡3
− 1)3
∙ 3𝑡2
𝑑𝑡 və 𝑡 =
3 4
𝑥 + 1-dir.
3 4
𝑥 + 1
𝑥
𝑑𝑥 =
𝑡
𝑡3 − 1 2
∙ 12𝑡2
∙ 𝑡3
− 1 3
𝑑𝑡 = 12 𝑡6
− 𝑡3
𝑑𝑡 =
=
12
7
𝑡7 − 3𝑡4 + 𝐶 =
12
7
4
𝑥 + 1
7
3 − 3 4
𝑥 + 1
4
3 + 𝑐

31. SƏRBƏST HƏLL ETMƏK ÜÇÜN ÇALIŞMALAR:
1.
𝑥2𝑑𝑥
𝑥−1 2 𝑥+1
2.
𝑑𝑥
(𝑥−1)(𝑥2−𝑥+1)
3.
𝑑𝑥
𝑥2−𝑥−2
4.
𝑑𝑥
𝑥3−𝑥2
5.
𝑑𝑥
𝑥3+1
6.
𝑑𝑥
𝑥 𝑥+1 2
7.
𝑥𝑑𝑥
𝑥3−1
8.
𝑑𝑥
(𝑥2−1)(𝑥+2)
9.
𝑥2𝑑𝑥
1−𝑥 3 10.
𝑑𝑥
𝑥3+𝑥

32. 11.
𝑑𝑥
6
𝑥5−
4
𝑥3
12.
𝑥3𝑑𝑥
𝑥−1
13.
𝑥𝑑𝑥
3
𝑥+1
14.
𝑥𝑑𝑥
2 𝑥+3
15.
𝑑𝑥
3+5 cos 𝑥
16. 𝑠𝑖𝑛 3 𝑥𝑑𝑥
17. 𝑠𝑖𝑛3
𝑥 𝑐𝑜𝑠2
𝑥𝑑𝑥 8. 𝑥 𝑠𝑖𝑛2
𝑥2
𝑑𝑥 19. 𝑐𝑜𝑠2
3𝑥 𝑠𝑖𝑛4
3𝑥 𝑑𝑥
20.
𝑑𝑥
𝑐𝑜𝑠2𝑥
21. sin 3𝑥 cos 5𝑥 𝑑𝑥 22. sin 10𝑥 sin 15𝑥 𝑑𝑥
23.
cos 𝑥𝑑𝑥
1+cos 𝑥
24. sin 9𝑥 sin 𝑥 𝑑𝑥 25. 𝑡𝑔2 (5𝑥)𝑑𝑥
26. 𝑡𝑔4 𝑥𝑑𝑥