instegration basic notes of class 12th h

integration notes

1. INTEGRATION

2. INTEGRATION
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4. INTEGRATION
●What is Integration
●Integration Notation
●Types of Integrations
●Constant of Integration
●Basic Integration Formulae
●Properties of Indefinite Integration
●Integration by Substitution
●Integration by Parts
●Some Standard Integration
●Definite Integration
●Geometrical Meaning of Integration
INTEGRATION BASIC

5. INTEGRATION
JEE(MAIN) PYQs Using INTEGRATION

6. INTEGRATION
What is Integration

7. INTEGRATION
Indefinite integration is the reverse process of
differentiation
Hence it is also called as antiderivative
What is Integration

8. INTEGRATION
Integration Notation
Integration is denoted by an integral sign ∫.
y = ∫f(x) dx+ = F(x) + c Constant of Integration
Integrand
Variable of
integration
F ’(x) also = f(x)
(First derivative)

9. INTEGRATION
Types of Integrations
Definite Integration
Indefinite Integration

10. INTEGRATION
Indefinite Integration as The Reverse Process of Differentiation
Derivative
Integral
Sin (x) + CCos (x)
d
dx
{F(x)} = f(x) then, ∫ f(x) dx = F(x) + cIf

11. INTEGRATION
e.g.
Similarly,
d
dx
(x
2
) = 2xso∫ 2x dx =x
2
+ c
d
dx
(sinx) = cos xso∫ cos x dx =sin x + c
Why c?
Indefinite Integration as The Reverse Process of Differentiation

12. INTEGRATION
d
dx
(x
2
) = 2x
d
dx
(x
2
+ 1) = 2x
d
dx
(x
2
+ 2) = 2x
.
.
.
.
∫ 2x dx = x
2
+ c
c = 0
c = 1
c = 2
.
.
where,
c = constant of integration
Thus, the general value
of ∫ f(x) dx= F(x) + c
d
dx
(x
2
+ c) = 2x
Constant of Integration

13. INTEGRATION
Constant of Integration: Graphical Approach

14. INTEGRATION
Q. What is Antiderivative of x
5
?
A
B
C
D
x
6
+c
5x
4
+c

15. INTEGRATION
Solution:

16. INTEGRATION
A
B
C
D
x
6
+c
5x
4
+c
Q. What is Antiderivative of x
5
?

17. INTEGRATION
Q. What is ?
A
B
C
D
x
4
+sin(x)+c
x
4
-sin(x)+c
12x
2
+sin(x)+c
12x
2
-sin(x)+c

18. INTEGRATION
Solution:

19. INTEGRATION
Q. What is ?
A
B
C
D
x
4
+sin(x)+c
x
4
-sin(x)+c
12x
2
+sin(x)+c
12x
2
-sin(x)+c

20. INTEGRATION
=x + cdx
∫
x
n
dx=
x
n+1
n+ 1
+ c(n ≠ –1)
d
dx
(x)=1. 1
d
dx
x
n+1
n+ 1
=2.
x
n
∫
3. =
x
1
x
dx
1
=log x+c
d
dx
(log x )
∫
Basic Integration Formulae

21. INTEGRATION
d
dx
e
x
=e
x
e
x
dx=e
x
+ c
4.
∫
d
dx
a
x
log a
=a
x
∫
a
x
dx=
a
x
log a
+ c
5.
Basic Integration Formulae

22. INTEGRATION
d
dx
(–cos x) =sin x
∫
sin x dx=–cos x+ c
d
dx
(sin x)= cos x
∫
dxcos x =sin x+c
d
dx
(tan x)
∫
dxsec
2
x=tan x+c
=sec
2
x
7.
8.
6.
Basic Integration Formulae

23. INTEGRATION
d
dx
(–cot x) =cosec
2
x
∫
cosec
2
x dx=–cot x+ c
9.
d
dx
sec x=sec x tan x
∫
sec x tan x dx=sec x+ c
10.
d
dx
(–cosec x) =cosec x cotx
∫
cosec x cot x dx=–cosec x+ c
11.
Basic Integration Formulae

24. INTEGRATION
Basic Integration Formulae: Summary
6.
7.
8.
9.
10.
11.

25. INTEGRATION
Properties of Indefinite Integration
∫ k f(x) dx = k ∫ f(x)dx
where k is a constant2.
1.∫ (f
1(x) + f
2(x)) dx =∫f
1(x) dx+∫ f
2(x) dx
3.

26. INTEGRATION
Q.

27. INTEGRATION
Solution:

28. INTEGRATION
Q.

29. INTEGRATION
Solution:

30. INTEGRATION
Q.

31. INTEGRATION
Solution:

32. INTEGRATION
Q.

33. INTEGRATION
Solution:

34. INTEGRATION
Q. Write the value of

35. INTEGRATION
Solution:

36. INTEGRATION
Q.

37. INTEGRATION
Solution:

38. INTEGRATION
Q. i) ∫ tan
2
x dx
ii) ∫ cot
2
x dx

39. INTEGRATION
Solution:
We know(1 + tan
2
x = sec
2
x)
tan
2
x= sec
2
x–1∴
∫(sec
2
x–1)dx∴I =
tan x–x+ c=
∫ tan
2
xdxi)
∫ cot
2
xdxii)
We know(1 + cot
2
x = cosec
2
x)
cot
2
x= cosec
2
x–1∴
∫(cosec
2
x –1)dx∴I =
–cot x–x+ c=

40. INTEGRATION
Q.∫
cos
2
x
4 –5 sin x
dx

41. INTEGRATION
Solution:
∫
cos
2
x
4 –5 sin x
dx
∫
4
cos
2
x
–
5 sin x
cos
2
x
dx=
∫
(4 sec
2
x –5 sec xtan x) dx=
We know ∫sec
2
xdx =tan x
∫&sec xtan xdx = sec x
= 4 tan x–5 sec x+ c

42. INTEGRATION
tan x cot x + C
tan x -cot 2x + C
tan x + cot x + C
tan x -cot x + C
Q.
A
B
D
C
A
B
D
C
D

43. INTEGRATION
Solution:

44. INTEGRATION
tan x cot x + C
tan x -cot 2x + C
tan x + cot x + C
tan x -cot x + C
Q.
A
B
D
C
A
B
D
C
D

45. INTEGRATION
Q.∫
x
3
–x
2
+ x–1
x–1
dx

46. INTEGRATION
∫
(x–1) (x
2
+ 1)
(x–1)
dx=
=
3
x
3
+ x + c
∫
(x
2
+ 1)dx=
Solution:

47. INTEGRATION
Q. Evaluate

48. INTEGRATION
Solution:

49. INTEGRATION
METHOD OF SUBSTITUTION

50. INTEGRATION
Steps for Integration by Substitution
1.Choose an appropriate function to substitute whose
derivative will replace the other terms of the Integral.
2.Determine the value of dx.
3.Substitute the integral
4.Integrate the resulting function.
5.Return to the initial variable.

51. INTEGRATION
I= ×g′ (x) dxf(g(x) )∫
Put g(x) = t
g′ (x) dx=dt
Differentiate w.r.t. x
I=f(t) dt∫∴

52. INTEGRATION
Q. Integrate: ∫ e
x
sin(e
x
)dx

53. INTEGRATION

54. INTEGRATION
Q. Integrate: ∫ 2xsin(x
2
+ 1)dx

55. INTEGRATION
We know that the derivative of (x
2
+ 1) = 2x.
Hence, let’s substitute (x
2
+ 1) = t,
so that 2x.dx = dt.
Therefore,

56. INTEGRATION
Q. Integrate:

57. INTEGRATION

58. INTEGRATION
Q. Integrate the function :

59. INTEGRATION
Solution:

60. INTEGRATION
Q.Evaluate

61. INTEGRATION
Solution:

62. INTEGRATION
Instead of solving entire sum,
just remember the shortcuts
∫f
n
(x) f
′
(x) dx1)
Special Cases of Substitution

63. INTEGRATION
∫t
n
dtI =
=
1
n+ 1
f
n+1
(x) + c
Put f (x) t=
∫f
n
(x) f
′
(x) dx
1)
f
′
(x)dx dt=
∫f
n
(x) f
′
(x)dx=
1
n + 1
f
n+ 1
(x) + c
Special Cases of Substitution
Shortcut

64. INTEGRATION
Q. Integrate: ∫5sin
4
x cosx dx

65. INTEGRATION

66. INTEGRATION
f
′
(x)
f(x)∫
dx2)
Special Cases of Substitution

67. INTEGRATION
Put f (x) t=
f
′
(x) dx dt=
1
t∫
dtI=
lnt + c=
lnf(x) +c=
∫
f
′
(x)
f(x)
lndx=f (x) + c
f
′
(x)
f(x)∫
dx2)
Special Cases of Substitution
Shortcut

68. INTEGRATION
Q.Integrate the function :

69. INTEGRATION
Solution:

70. INTEGRATION
∫
f′(x)
f(x)
dx3)
√
Special Cases of Substitution

71. INTEGRATION
= 2√t+2
=2f(x) + c√
Put f (x)t=
f′(x) dx dt=
1
t∫
dtI =
√
∫
f′(x)
f(x)
dx3)
√
Special Cases of Substitution
∫
f′(x)
f(x)
dx
√
Shortcut

72. INTEGRATION
Q.Integrate:∫
cosx
sinx
dx
√

73. INTEGRATION

74. INTEGRATION
d
dx
(sin
–1
x)=
1
1 –x
2
√
∫
1
1 –x
2
√
dx
=sin
–1
x+c
d
dx
(tan
–1
x)=
1
1 +x
2
∫
1
1 +x
2
dx
=tan
–1
x+c
Some Other Integrals

75. INTEGRATION
Some Other Integrals

76. INTEGRATION
Some Other Integrals

77. INTEGRATION
Q.Integrate:

78. INTEGRATION
Solution:

79. INTEGRATION
Q.Integrate:

80. INTEGRATION
Solution:

81. INTEGRATION
INTEGRATION BY PARTS

82. INTEGRATION
I
L
A
T
E
INVERSE
LOGARITHMIC
ALGEBRAIC
TRIGONOMETRIC
EXPONENTIAL
⮚To solve Integration by Parts we
make use of ILATE rule
∫
f(x) g(x)= –f(x) ∫
g(x)dx
d
dx
f(x)×∫
g(x)dxdx∫

83. INTEGRATION
Q.∫
xsin xdx

84. INTEGRATION
Algebraic Trigonometric
Solution:

85. INTEGRATION
Q.

86. INTEGRATION
=ln xf (x)
g (x)=1
∫ln x×1 dx
=xln x–x+ c
=ln (x) x –∫x
1
×xdx
=x ln (x) –∫1dx
Solution:

87. INTEGRATION
Q.∫
x
2
cos2xdx

88. INTEGRATION
f(x)=x
2
g(x)=cos2x
I
sin 2x
=x
2
2
–∫
2x×
sin 2x
2
dx
=x
2
sin 2x
2
–
x (–cos 2x)
2
–∫
(–cos 2x)
2
dx
sin 2x sin 2x
=x
2
2
–
–x cos 2x
2
+
4
=x
2
sin 2x
2
+
x cos 2x
2
–
sin 2x
4
+c
Solution:

89. INTEGRATION
Some standard integrals
dx
x
2
+ a
2∫
1
a
tan
–1
x
a
+c=
dx
a
2
–x
2∫
1
2a
ln
a+ x
a–x
+c=
dx
x
2
–a
2∫
=
1
2a
ln +c
x -a
x+ a
•
•
•
dx
a
2
–x
2∫
dx
x
2
–a
2∫
=
_

90. INTEGRATION
=ln+ cx +
•∫
dx
x
2
+ a
2
√
•
∫
dx
a
2
–x
2√
sin
–1
x
a
+c=
x
2
+ a
2√
=ln+ cx +
•∫
dx
x
2
-a
2
√
x
2
-a
2
√
Some standard integrals

91. INTEGRATION
Q.Evaluate:

92. INTEGRATION

93. INTEGRATION
Q.Evaluate

94. INTEGRATION
Solution:

95. INTEGRATION
Definite Integration

96. INTEGRATION
Geometrical Meaning of Integration
Definite Integrations
Indefinite Integrations
Unbounded Area
Bounded Area

97. INTEGRATION
Q.

98. INTEGRATION
Solution:

99. INTEGRATION
Q.
B
D
C
A

100. INTEGRATION
Solution:

101. INTEGRATION
Q.
B
D
C
A

102. INTEGRATION
JEE MAINS & ADVANCED COURSE
➔FoundationSessionsforstarters
➔CompletePYQ’s(2015-2023)
➔NTA+CengageCHAPTERWISEQuestions
➔MyHANDWRITTENNotes
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WE DO NOT SELL ANY COURSES
For FREE & Focused JEE MATERIAL, CLICK to Join TELEGRAM :
t.me/mathematicallyinclined