Chapter 7 Class 12

Integration Formula Sheet

 + 𝐶, 𝑛 ≠ −1. 𝑃𝑎𝑟𝑡𝑖𝑐𝑢𝑙𝑎𝑟𝑙𝑦 , ∫ 𝑑𝑥 = 𝑥 +
Basic Formulae

1. ∫ 𝑥𝑛 𝑑𝑥 = 𝑐

2. ∫ -----𝑑𝑥 = sin x + C

3. ∫ 𝑠𝑖𝑛 𝑥 𝑑𝑥 = ---- + C

4. ∫ ---- 𝑥 𝑑𝑥 = tan x + c

5. ∫ 𝑐𝑜𝑠𝑒𝑐2 𝑥 𝑑𝑥 = - - - - + c

6. ∫ ------ 𝑑𝑥 = sec x + c

7. ∫ 𝑐𝑜𝑠𝑒𝑐 𝑥 𝑐𝑜𝑡 𝑥 𝑑𝑥 =- - - - + c

𝑑𝑥
8. ∫ 1−
= ---- + c Questions in
𝑥2
Ex 7.2 and Ex 7.3
𝑑𝑥
9. ∫ = – cos-1 x + c

𝑑𝑥
10. ∫ = ---- + c
1 +
𝑥2
 𝑑𝑥
11. ∫
= – cot-1 x +
c
-------

12. ∫ 𝑒 𝑥 𝑑𝑥 = --- + c

13. ∫ 𝑎𝑥𝑑𝑥 = +c

𝑑𝑥
14. ∫ = ----- + c
𝑥 𝑥2 −

𝑑𝑥
15. ∫ = ------ + c
𝑥 𝑥2 −

16. ∫ 𝑑𝑥 = log 𝑥 + c

17. ∫ --- 𝑑𝑥 = log sec x + c

18. ∫ 𝑐𝑜𝑡 𝑥 𝑑𝑥 = log ------ + c

19. ∫----- 𝑑𝑥 = log sec x + tan 𝑥+ c

20. ∫ 𝑐𝑜𝑠𝑒𝑐 𝑥 𝑑𝑥 = log -------------------+ c

Integrals of some special functions

𝑑𝑥
∫ +𝑐
1

𝑥2 − 𝑎2 log
1.
2𝑎
=

𝑑𝑥
∫ +𝑐
1

--------- log 𝑎+
2.
= 2𝑎 𝑥
𝑎−

𝑑𝑥 1 −1 𝑥
∫ ta +𝑐
n
3.
𝑥2 + 𝑎2 --
= -- Questions in
Ex 7.4
𝑑𝑥
∫ = log -- + --------- + C
𝑥2 −
4.

𝑑𝑥
5. ∫ --------
= sin-1 + 𝑐
𝑥

𝑎
 𝑑𝑥
∫
𝑥2 +
6. = log ----- +C
Integrals by partial fractions

,𝑎 ≠
𝑝𝑥 + �
𝑞 (𝑥 − 𝑏)= ------
�
𝑥−
1. � + ------
�
𝑎 b
�
𝑥−
�𝑎
= �
𝑥−� 2
2.
+ 𝑎

𝑝𝑥2 + 𝑞𝑥 𝐴 𝐵 𝐶 Questions in
𝑥 −+𝑎𝑟𝑥 𝑥−
3. = 𝑥 − 𝑎+ 𝑥 − + 𝑥 −
−𝑏 𝑐 𝑏 𝑐
Ex 7.5
𝑝𝑥2 + 𝑞𝑥 �
+𝑟 𝑥−
�𝑎 2 𝑥
� �
𝑥−𝑎
4. = � + + �
𝑥−𝑎 2 𝑥 −𝑏
𝑝𝑥2 + 𝑞𝑥 + 𝑟 � -------
𝑥− 𝑎 𝑥2 + 𝑏𝑥 𝑥 − 𝑎 𝑥2 + 𝑏𝑥
5. = +
+𝑐
�

Where 𝑥 2 + bx + c can not be factorised further.

 Integration by parts

1. ∫ 𝒙 � 𝒙 𝒅𝒙 = ∫ 𝒅𝒙 − 𝒅𝒙
𝒇 � 𝒇 𝒙𝒈 𝒙∫ �𝒙 ∫ 𝒈 𝒙
� 𝒅𝒙

To decide first function. We use

I → Inverse (Example 𝑠𝑖𝑛−1 𝑥)
L → Log (Example log 𝑥)
A → Algebra (Example x2, x3)
Questions in
Ex 7.6
T → Trignometry (Example sin2 x)
E → Exponential (Example ex)

2. ∫𝑒𝑥 𝑓𝑥 + 𝑓′(𝑥) dx = ∫𝑒𝑥 f(x) dx + C

 Other Special Integrals

𝒙𝟐 −
1. ∫ 𝒂𝟐 𝒅𝒙 =
\

𝒙𝟐 +
2. ∫ 𝒅𝒙 =

𝒂𝟐 −
3. ∫ 𝒙𝟐 𝒅𝒙 =

Questions in
Ex 7.7
Area as a sum

∫ 𝑥 𝑑
𝑓 𝑥 1
𝑙𝑖𝑚 𝑎 + 𝑎+ + 𝑓𝑎 + …+
= 𝑏− (𝑓 𝑓 ℎ 2ℎ
𝑎 𝑛→∞ 𝑛

Questions in
Ex 7.8
Properties of definite integration

�𝑥 𝑑𝑥 = �𝑡 𝑑𝑡 =
𝑏

∫𝑎 � ∫
𝑏
P0 :
�
��

�𝑥 𝑑𝑥 = − � 𝑑𝑥 . In
𝑏
� 𝑑𝑥 = 0
∫𝑎 � ∫
𝑎
� particular, ∫
𝑎
P1 :
�

�𝑥 𝑑𝑥 = � 𝑑𝑥 + � 𝑑𝑥
𝑏

∫𝑎 � ∫
𝑐
� ∫
𝑏
P2 :
�

�𝑥 𝑑𝑥 =
𝑏
𝑓 𝑎 + 𝑏 − 𝑥 𝑑𝑥.
∫𝑎 � ∫
𝑏
P3 :

�𝑥 𝑑𝑥 = 𝑓 𝑎 − 𝑥 𝑑𝑥
𝑎

∫0 � ∫
𝑎
P4 :

P5 : ∫0 �𝑥 𝑑𝑥 = 𝑓 𝑥 𝑑𝑥 + 𝑓 2𝑎 − 𝑥 𝑑𝑥
2𝑎

𝑎 𝑎
� ∫ ∫

P : ∫2𝑎
𝑓 2
= { 𝑎
∫0 𝑓 𝑥 𝑑𝑥 , 𝑖𝑓 𝑓 2𝑎 − 𝑥 = 𝑓(𝑥)
𝑥
0
0, 𝑖𝑓 2𝑎 − = −𝑓(𝑥)
6

𝑓 𝑥

0
P : ∫𝑎 𝑥 −𝑥
� 𝑥 = { ∫ 𝑎 𝑑𝑥 , = 𝑓(𝑥)
� 2
𝑖𝑓 𝑓
𝑓
 −𝑎
0, 𝑖𝑓 = −𝑓(𝑥)
6

�
𝑥 �
𝑥
�

𝑥 𝑥
�

� �
� �

�
� Questions in
0 Ex 7.11

−𝑥