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Chapter 7 Class 12
Integration Formula Sheet
by teachoo.com
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Basic Formulae
1.
2.
𝑐𝑜𝑠 𝑥 𝑑𝑥 = sin x + C
3.
𝑠𝑖𝑛 𝑥 𝑑𝑥 = −cos x + C
4.
𝑠𝑒𝑐2 𝑥 𝑑𝑥 = tan x + c
5.
𝑐𝑜𝑠𝑒𝑐2 𝑥 𝑑𝑥 = −cot x + c
6.
𝑠𝑒𝑐 𝑥 𝑡𝑎𝑛 𝑥 𝑑𝑥 = sec x + c
7.
𝑐𝑜𝑠𝑒𝑐 𝑥 𝑐𝑜𝑡 𝑥 𝑑𝑥 = −cosec x + c
8.
9.
10.
𝑑𝑥 =
𝑥𝑛+1
𝑛+1
𝑥𝑛
𝑑𝑥
1 − 𝑥2
𝑑𝑥
1 − 𝑥2
𝑑𝑥
1 + 𝑥2
+ 𝐶, 𝑛 ≠ −1. 𝑃𝑎𝑟𝑡𝑖𝑐𝑢𝑙𝑎𝑟𝑙𝑦 , 𝑑𝑥 = 𝑥 + 𝑐
= sin-1 x + c
= – cos-1 x + c
= tan-1 x + c
Questions in
Ex 7.2 and Ex 7.3
11.
𝑑𝑥
1 + 𝑥2
12.
𝑒 𝑥 𝑑𝑥 = 𝑒 𝑥 + c
13.
𝑎 𝑥 𝑑𝑥
14.
15.
16.
= – cot-1 x + c
=
𝑑𝑥
𝑥 𝑥2 − 1
𝑑𝑥
𝑥 𝑥2 − 1
1
𝑑𝑥
𝑥
𝑎𝑥
log 𝑎
+c
= sec-1 x + c
= −cosec -1 x + c
= log 𝑥 + c
17.
𝑡𝑎𝑛 𝑥 𝑑𝑥 = log sec x + c
18.
𝑐𝑜𝑡 𝑥 𝑑𝑥 = log sin x + c
19.
𝑠𝑒𝑐 𝑥 𝑑𝑥 = log sec x + tan 𝑥 + c
20.
𝑐𝑜𝑠𝑒𝑐 𝑥 𝑑𝑥 = log cosec x − cot 𝑥 + c
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Integrals of some special functions
1.
𝑑𝑥
𝑥 2 − 𝑎2
2.
𝑑𝑥
𝑎2 − 𝑥 2
3.
𝑑𝑥
𝑥 2 + 𝑎2
4.
5.
6.
𝑑𝑥
𝑥2 − 𝑎
𝑑𝑥
𝑎2 − 𝑥
𝑑𝑥
𝑥2 + 𝑎
=
1
𝑥−𝑎
log
2𝑎
𝑥+𝑎
+𝑐
=
1
𝑎+𝑥
log
2𝑎
𝑎−𝑥
+𝑐
=
1
𝑥
−1
tan
𝑎
𝑎
+𝑐
= log x +
2
𝑥 2 − 𝑎2 + C
𝑥
𝑎
= sin-1 + 𝑐
2
= log x +
2
𝑥 2 + 𝑎2 + C
Questions in
Ex 7.4
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Integrals by partial fractions
1.
𝑝𝑥 + 𝑞
𝑥 − 𝑎 (𝑥 − 𝑏)
2.
𝑝𝑥 + 𝑞
𝑥−𝑎 2
3.
𝑝𝑥 2 + 𝑞𝑥 + 𝑟
𝑥−𝑎 𝑥−𝑏 𝑥−𝑐
4.
𝑝𝑥 2 + 𝑞𝑥 + 𝑟
𝑥−𝑎 2 𝑥−𝑏
5.
𝑝𝑥 2 + 𝑞𝑥 + 𝑟
𝑥 − 𝑎 𝑥 2 + 𝑏𝑥 + 𝑐
=
=
𝐴
𝑥−𝑎
𝐴
𝑥−𝑎
+
=
+
𝐵
,
𝑥−𝑏
𝑎≠b
𝐵
𝑥−𝑎 2
𝐴
𝑥−𝑎
=
𝐴
𝑥−𝑎
=
+
+
𝐵
𝑥−𝑏
𝐵
𝑥−𝑎 2
𝐴
𝑥−𝑎
+
+
+
𝐶
𝑥−𝑐
𝐶
𝑥−𝑏
𝐵𝑥 + 𝐶
𝑥 2 + 𝑏𝑥 + 𝑐
Where 𝑥 2 + bx + c can not be factorised further.
Questions in
Ex 7.5
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Integration by parts
1.
𝒇 𝒙 𝒈 𝒙 𝒅𝒙 = 𝒇 𝒙
𝒈 𝒙 𝒅𝒙 −
𝒇′ 𝒙
𝒈 𝒙 𝒅𝒙 𝒅𝒙
To decide first function. We use
I → Inverse (Example 𝑠𝑖𝑛−1 𝑥)
L → Log (Example log 𝑥)
A → Algebra (Example x2, x3)
T → Trignometry (Example sin2 x)
E → Exponential (Example ex)
2. 𝑒𝑥 𝑓 𝑥 + 𝑓′(𝑥) dx = 𝑒𝑥 f(x) dx + C
Questions in
Ex 7.6
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Other Special Integrals
log 𝑥 + 𝑥 2 − 𝑎2 + C
+
𝑎2
2
log 𝑥 + 𝑥 2 + 𝑎2 + C
𝑎2 − 𝑥 2 +
𝑎2
2
sin1 + C
𝒅𝒙 =
𝑥2
𝒅𝒙 =
𝑥
2
𝑥2
𝒂𝟐 − 𝒙𝟐 𝒅𝒙 =
𝑥
2
1.
− 𝒂𝟐
2.
𝒙𝟐
+ 𝒂𝟐
3.
−
𝑎2
2
𝑥
2
𝒙𝟐
−
𝑎2
+
𝑎2
𝑥
𝑎
Questions in
Ex 7.7
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Area as a sum
𝑓 𝑥 𝑑𝑥
= 𝑏−𝑎
1
𝑙𝑖𝑚
𝑛→∞ 𝑛
𝑓 𝑎 + 𝑓 𝑎 + ℎ + 𝑓 𝑎 + 2ℎ … +
Questions in
Ex 7.8
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Properties of definite integration
P0 :
𝑏
𝑓
𝑎
𝑥 𝑑𝑥 =
P1 :
𝑏
𝑓
𝑎
𝑥 𝑑𝑥 = −
P2 :
𝑏
𝑓
𝑎
P3 :
𝑏
𝑓
𝑎
P4 :
𝑎
𝑓
0
P5 :
2𝑎
𝑓
0
𝑏
𝑓
𝑎
𝑡 𝑑𝑡 =
𝑎
𝑓
𝑏
𝑥 𝑑𝑥 . In particular,
𝑥 𝑑𝑥 =
𝑐
𝑓
𝑎
𝑥 𝑑𝑥 +
𝑥 𝑑𝑥 =
𝑏
𝑓
𝑎
𝑎 + 𝑏 − 𝑥 𝑑𝑥.
𝑎
𝑓
0
𝑎 − 𝑥 𝑑𝑥
𝑥 𝑑𝑥 =
𝑥 𝑑𝑥 =
𝑎
𝑓
0
𝑥 𝑑𝑥 +
𝑏
𝑓
𝑐
𝑎
𝑓
𝑎
𝑥 𝑑𝑥 = 0
𝑥 𝑑𝑥
Questions in
Ex 7.11
𝑎
𝑓
0
2𝑎 − 𝑥 𝑑𝑥
2
𝑥 =
0,
𝑎
𝑓
0
𝑥 𝑑𝑥 , 𝑖𝑓 𝑓 2𝑎 − 𝑥 = 𝑓(𝑥)
𝑖𝑓 𝑓 2𝑎 − 𝑥 = −𝑓(𝑥)
2
𝑎
P6 : −𝑎 𝑓 𝑥 =
0,
𝑎
𝑓
0
𝑥 𝑑𝑥 , 𝑖𝑓 𝑓 −𝑥 = 𝑓(𝑥)
𝑖𝑓 𝑓 −𝑥 = −𝑓(𝑥)
P6 :
2𝑎
𝑓
0