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CHAPTER 7
INTEGRALS
POINTS TO REMEMBER
Integration is the reverse process of Differentiation
...
From geometrical point of view an indefinite integral is collection of family
of curves each of which is obtained by translating one of the curves
parallel to itself upwards or downwards along y-axis
...
STANDARD FORMULAE
1
...
x n 1
c
x dx n 1
log x c
n
n 1
n –1
ax b n 1
c
n 1 a
n
ax b dx
1
log ax b c
a
3
...
tan x
...
– cos x c
...
– log cos x c log sec x c
...
cot x dx
8
...
dx
2
cosec x cot x dx
11
...
cosec x dx
13
...
15
...
1 x
17
...
19
...
2
1 x
2
2
x
2
a
2
14
...
–1
log
2a
dx
1
x c , x 1
...
a x
log
2a
dx
x
a dx
x c, x 1
...
dx sec
1
a
– cosec x c
...
tan x
...
dx tan x c
...
dx tan
2
sec
9
...
10
...
log sin x c
...
x a
–1
x
c
...
21
...
23
...
2
dx log x
dx log x
x
2
25
...
2
a
2
2
2
a x
a x dx
x
a
24
...
–1
c
...
a
2
2
log x
a x
log x
x
2
c
...
2
RULES OF INTEGRATION
1
...
f x dx k f x dx
...
k f x g x dx k f x dx k g x dx
...
2
...
n
f x f ´x dx
f x n 1
c
...
f ´ x
f x
n
f x n 1
dx
c
...
g x dx
f x
...
g x dx dx
...
a
DEFINITE INTEGRAL AS A LIMIT OF SUMS
...
f a n 1 h
h0
b a
h
b
...
a
c
b
f x dx
a
f t dt
...
a
a
b
4
...
b
b
3
...
(i)
a
XII – Maths
c
b
f x dx
a
f a b x dx
...
0
a
5
...
–a
a
6
...
a
2a
7
...
3
...
2
...
1
1
x
x8
8
6
...
e
1
5
...
1
7
...
4 3 cos x
8
...
cos 2x 2sin2 x
dx
...
2
11
...
8 x
dx
...
1 sin2 x dx
...
2
7
sin
x dx
...
65
d
f x dx
...
15
...
e
17
...
x
dx
...
x 1
21
...
sec x
...
2
1
dx
...
16
...
ex
a x dx
...
13
...
x
dx
...
22
...
24
...
26
...
28
...
30
...
1
ax ax dx
...
cot x
...
27
...
29
...
sin x
x 1
33
...
0 x dx
x e 1 e x 1
dx
...
31
...
2
XII – Maths
where [ ] is greatest integer function
...
2
2
0 x dx
where [ ] is greatest integer function
...
a f x f a b x dx
...
If
40
...
39
...
a f x dx b f x dx
...
1
a
1
0 1 x 2
, then what is value of a
...
(i)
x cosec
tan–1x 2
1 x
4
dx
...
cos x a
(iii)
sin x a sin x b dx
...
(v)
cos x cos 2x cos 3x dx
...
sin x cos x
2
2
2
dx
...
(xii)
sin x cos x
67
5
3
x dx
...
1
dx
...
sin 2x
XII – Maths
42
...
1
*(ii)
x 6 log x
(iii)
1 x x
(v)
(vii)
(ix)
(xi)
43
...
...
dx
...
2
3x 2
x
2
x 1 dx
...
(iii)
cos
XII – Maths
(vi)
7
1
...
cos 2
68
(xii)
x
x
1
9 8x x
2
sin x
dx
...
2
dx
...
sec x 1 dx
...
x 1
x 1 x 2 x 3 dx
...
(vi)
...
(xii)
tan x dx
...
dx
sin x 1 2 cos x
...
1
dx
...
cos
sin
(vi)
1 sin 2x
dx
...
(xi)
e
x
(xii)
log log x log x
2
x
3
(viii)
x dx
...
e
x
x
(x)
e
3
x dx
...
x dx
...
2x 2
x 2
1
x 12
dx
...
1 cos 2x
1
dx
...
6x 5
x
2
2
6 x x dx
...
x 3
x
2
4x 3 dx
...
Evaluate the following definite integrals :
4
(i)
sin x cos x
9 16 sin 2x
2
dx
...
x
0
1 x
1 x
1 2
2
2
(iv)
dx
...
2
(v)
sin
0
2
sin 2x
4
4
dx
...
4x 3
2
(vii)
x sin x
1 cos x dx
...
Evaluate :
3
(i)
x 1 x 2 x 3 dx
...
4
(iii)
2
log 1 tan x dx
...
(iv)
0
0
(v)
x sin x
1 cos
2
0
dx
...
2
(vii)
x sin x cos x
sin
0
4
4
(viii)
a
x
2
0
47
...
x cos x
2
2
dx
...
sec x cosec x
(iv)
e
0
a
dx
...
1 x2
e
cos x
cos x
e
a x
cos x
dx
...
a x
XII – Maths
1
48
...
0
log x log sin x
49
...
dx
...
52
...
b
4
53
...
sin x dx
...
dx
...
4
a
55
...
56
...
57
...
58
...
59
...
1
LONG ANSWER TYPE QUESTIONS (6 MARKS)
60
...
(ii)
x
2x
dx
x 1 x
3
x 1 x 3
2
dx
(iv)
x
x
4
x
4
4
2
dx
4
dx
– 16
2
(v)
cot x dx
...
x tan
x
2 2
1
dx
...
Evaluate the following integrals as limit of sums :
4
(i)
2
2x 1 dx
...
0
73
XII – Maths
3
2
3x
(iii)
4
2x 4 dx
...
0
1
5
x
(v)
2
2
3 x dx
...
Evaluate
1
cot
(i)
1
1
x x 2 dx
0
dx
sin x 2 cos x 2 sin x cos x
(ii)
1
(iii)
log 1 x
0
1 x2
63
...
65
...
2
dx
2 log sin x
(iv)
– log sin 2x dx
...
5 cos2 4 sin d
...
e
2x
cos 3x dx
...
2
...
2e – 2
2
3
...
4
...
log8 9
16
5
...
log | log (log x) | + c
XII – Maths
74
8
...
0
9
...
0
11
...
f(x) + c
13
...
2 32 2
32
x x 1 c
3
3
15
...
e
a
17
...
2
x 13 2 2 x 11 2 c
...
log x 1
20
...
x cos2 + c
22
...
cos
23
...
log cos x sin
c
sin
25
...
x4
1
3x 2
2
3 log x c
...
x 1
2
2
x
x
log e a c
c
27
...
3
28
...
2 log |sec x/2| + c
...
1
log x e e x c
...
x log x 2 c
32
...
0
34
...
36
...
–1
38
...
1
40
...
2
a
(i)
1
1
log cosec tan1 x 2 2 c
...
1 2
1
x x x 2 1 log x x 2 1 c
...
1
(v)
log
12x 6 sin 2x 3 sin 4x 2 sin 6x c
...
32
2
2
6
(viii)
4
cot6 x
cot x
c
...
5
1
(ix)
a2
b
2
2
2
2
2
c
...
sin a c
...
: Take sec2 x as numerator]
(xi)
(xii)
42
...
sin–1 (sin x – cos x) + c
...
[Hint : put x2 = t]
C
[Hint : put log x = t]
3 log x 2
1
log
5
5 – 1 2x
c
5 1 2x
(iv)
x 4
sin1
c
...
log sin x
cos
Hint :
5
(vii)
(viii)
(ix)
2
log 3 x 2 x 1
6
4x x
sin x
sin x
11
tan
1
3 2
x 3 log x
2
2
2
1
2
2
sin x sin
sin x
3 x 1
c
2
6x 12 2 3 tan
4 sin
2
sin x sin c
1
x 3
c
3
x 2
c
2
77
XII – Maths
(x)
1
1 x x
3
2 2
1
3
2x 1 1 x x 2
8
5
sin
1
16
1
2
x x x 1
7
2
c
1
2
2 3
x x 1
log x
2
8
3
2
(xi)
x 2
(xii)
log cos x
x 1
2 x 1
c
5
1
2
cos x cos x c
2
[Hint : Multiply and divide by
43
...
3
log x 3
10
4
log x 2
15
x 4 log
x 2 2
1
x 1 c
6
c
x 1
(vi)
x
2
3
tan
x
1 x
3 tan c
2
3
1
[Hint : put x2 = t]
(vii)
XII – Maths
2
17
log 2x 1
1
log x
17
2
4
1
34
78
tan
1
x
2
c
sec x 1 ]
(viii)
1
log 1 cos x
2
1
log 1 cos x
6
2
log 1 2 cos x c
3
[Hint : Multiply Nr and Dr by sin x and put cos x = t]
(ix)
1
1 sin x
log
8
(x)
1
1 sin x
log
2
(xi)
x
x
1
2
1
tan
44
...
sec2 x and take sec x as first function]
(ii)
(iii)
e
ax
2
a b
(iv)
2x tan
2
1
a cos bx
3x
1
c b sin bx c c1
log 1 9x
2
c
[Hint : put 3x = tan ]
3
(v)
(vi)
2 x sin x cos x c
3
x 4 1
x
x
1
tan x –
c
...
2
x a
2ax x
e
(viii)
2
2
a
2
1
sin
2
x
c
...
x 1
(xi)
ex tan x + c
...
[Hint : put log x = t x = et ]
log x
(xiii)
2 6 x x
2 32
25
2x 1
2
1 2x 1
8
6 x x
sin
c
5
8
4
(xiv)
(xv)
1
x 2 x 2 9 3 log x
2
x
2
9 c
2
2
x
2
4x 3
3
1
log x 2
3
2
x 2
2
x
2
x
2
4x 3
4x 3 c
2
(xvi)
x 2
2
1
45
...
(ii)
20
80
–
4
x
2
4x 8 c
(iii)
(v)
–
4
1
...
(i)
(iii)
log 2
...
5
2
x
sin x
Hint :
dx
...
(ii)
log 2
...
2
2
...
Hint :
(vii)
8
1
(v)
1
25
/2
...
5 – 10 log
(i)
2
(vi)
46
...
16
Hint : Use
2
...
(ii)
12
0
...
2
/2
...
1
2
49
...
50
...
51
...
2
52
...
2 2
54
...
log |1 + sin x| + c
56
...
log |cos x + sin x| + c
58
...
a c x b c x C
...
XII – Maths
1
1
1 2
3
x
3 2
x x2 c
1
2
log 1 2 c
x 3
82
(iv)
sin x x cos x
c
x sin x cos x
(v)
x a tan1 x ax c
(vi)
2 sin1
(vii)
0
(viii)
2
(ix)
60
...
x 4 log x
5
log x 1
4
3
log x 1
4
log x
2
1
1
tan
–1
x c
...
1
–1
x
c
...
x
c
...
1
2 2
tan
1
x2
1
2x
1
4 2
83
log
x
x
2
2
2x 1
2x 1
c
XII – Maths
(vii)
/8
...
(iii)
26
...
(ii)
127 e
...
8
(iv)
62
...
2
141
(v)
8
...
2
2
1
tan x x
log
c
5
2 tan x 1
63
...
6
2
3
64
...
1
1
sec x tan x log sec x tan x c
...
e 2x
2cos 3x 3 sin3x c
...
2 sin
84