INTEGRALS
tan
C
i o l u t i o n
Example30
(x) dr =0 iffis an
-a
function.
Solution Odd.
2a
Fample31 Jf() dr
=
2Jf(x) dr, iff(2a-)=
Solution (r).
Example 32Sin
sin" x dx
x+cos" x
Solution 4
7.3 EXERCISE
Short Answer (S.A.)
Verify the following:
1.
d
=x-log (2x+31+C
2x+3
2.
xSdr
=
+3x
log k?
+3x| +C
Evaluate the followingg
6log
3.
2)d
x+1
J 4logr hgr
163
164
MATHEMATICS
6.
l+cosx
+Cosx
x+SinxX
Sin x+COS x
1
tan' xsec' xdr
9.
V+sin xdr
8.
yx =2)
10.
(Hint: Put
12.
(Hint: Put x = z)
+sin 2x=dx
a+x
11.
a-x
13.
1+r
dt
14.
16-92
15.
3-2
16. 4
17.
V5-2x+x'd
18.
19.
Jds putr-t
20.
22
V2ax-xdr
21.
(cos Sx +cos 4x)
dx
1-2cos3x
sin dx
(1-)
sin° x +cos *
23.
dr
sin
x cos' x
INTEGRALS
25.
dr
COs x- cos 2x
1-cosx
-
165
dx
Hint: (Hint : Put x = sec 0)
lhuate the following as limit of sums:
J+3)dr
28.
Je d
Evaluate the following:
dx
30.
e+e
tan xdr
1+m tan' x
dr
31.
xdx
ir-1)(2-x)
32
dr
33
x sin xcos
34.
xd
0
1+r)1-x
(Hint:letr= sin6)
Long Answer (LA.)
rdr
35
37.
xdx
-12
36.
2x-1
38.
1+sin x
(r+ a Xx+b')
dx
(x-1)(r+2)(r-3)
166
MATHEMATICS
n'
40.
+rtx
Vatx
39.
42.
+cos.x*
(Hint:
Put
Je
cos'x d
x
=
a
tan')
5
41.
(-cosx)2
43.
44
tan xdr
(Hint: Put
tanx
=
P)
dx
(a cos x+b° sin* x)
(Hint: Divide Numerator and Denominator by costx)
45.
47.
46.
r log(1+2x)dr
xlogsin xdr
log(sin x+cosx)dx
Objective Type Questions
Choose the correct option from
48.
given four options in each of the Exercises from 48 to 0
cos2x-cOS 4D dr is equal to
COS X-Ccos
(A) 2(sinx +xcos8) +C
(C) 2(sinx + 2xcos6) +C
(B) 2(sinx - xcos6) +C
(D) 2(sinr -2x cose) +C
INTEGRALS 167
dx
is equal
cqual to
to
*
sinr-a) Sin(x-b)
(A) sin (
-a) log
Isin(x-b+C
sin(x-a)
C)cosec (b-a) log
.
4
sinx-0)+C
|sin(x - a)
(B) cosec (b - a) log
(D) sin (b- a) log
sin(x-a)+C
sin(x b)
sinx-a)c
sin(x - b)
tan x dr is equal to
(A)(r+1) tan" vx-vx+C
(B) xtan-V+C
(C) x-x tan" vx +C
D) - x +1)tan" vx +C
dx is equalto
(B)
e
(A) 4+C
C (1+
52.
(4x+1
(D) (1+
+C
dr is
C
equalto
+C
C)(1+4)*+C
10x
168
MATHEMATICS
dx
53.
If g+2(r? +1)
log|l
+r|+b
?
Ca1054.
(B) a= 0b-
D)ab
2
is equal to0
54.
x
-loglh-d+ C
(C)x
-logl1+ xl+C
(A)
56.
x+21+C, then
r4
Cx+2+14
(A)a10,b 5
55.
tan'x +log
(B) x+
-logl1-xl+C
(D) x+logl1+a+C
(X+Sin x a is equal to
1+cosx
(A) logll+ coszl+ C
(B) loglx+ sin x+ C
(C)x-tan+C
D) x.tan
+C
x dx
If
VI+
al+*)i+by1+x+C,then
(A) a
b=1
C)a
b=-1
(B)a
(D)a-
b=1
b b-1
INTEGRALS
d
169
is equal to
+cos2
(B) 2
(A)1
C) 3
(D) 4
C)2
D) 22-1)
-sin 2xdx is equal to
0
B) 2 (2+1)
(A) 22
59.
cosxe"dr is equal to
0
fI+3
60.
Fill in the blanks in each of the following Exercise
61.
62.
63.
dx =
1+4x
Sin x
then
8
a =
-dx
3+4cos x
The value ofsin'x cosx dx is
60 to 63.
ANSWERS 295
33. Rs 1920
36. B
37. A
40. A
D
41. A
44. B
B
A
B
.(1, o)
42. D
45. C
48. A
D
38. C
46. B
49. B
52. C
50. C
53. B
54. C
56. A
57. B
58. B
60. (3,34)
61. x+y =0
62. (-oa-1)
64. 2ab
7.3 EXERCISE
4tC
4.
-x+3logr+1+c
6
tanx tan
tan+C
7.
2
5
+C 8. x+c
3
9-2cos+2Sin*C
10.
11.-acos
12
3
4-log|l+|+
14. sin
+c
15. i
5. log|x +sin +c
.*
16. 3W+9-loglr+ +
9+
3x
296
MATHEMATICS
17.
2
s-2r+r +2log/x-I+s-2x+r|+e
18.log|-bgls'
+e
4
20.
2ar-
22.
sin 2x
+sin"
sin x +
.
lo23. tanx
c
-
logm
28. -1
30. m21
32. 2-1
33.
34.
X
a tan-btan
40.
3x
+ c
31. 7T
3
35. o
log
-
27.
sec
()+e
2
29. tan 'e
38.
cot x
25. 2 sinx +x+c
24.in
26.
tan' x+c
19.
-3
x-16 (x +2)
+c
c 37. T
39. xean
+C
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