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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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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