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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
teachoo.com
teachoo.com 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
teachoo.com 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
teachoo.com 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
teachoo.com 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
teachoo.com 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