Page 236 Integrals CHAPTER 7

CHAPTER 7
INTEGRALS

Sol :
OBJECTIVE QUESTIONS
Delhi 2012

7. is equal to
1. The integral is equal to :
(a) (b)
(a) (b) (c) 1 (d)
Sol : OD 2013
(c) (d)
Sol : OD 2024

8. is equal to

2. The value of d is (a)

(a) (b) (b)
(c)
(c) 0 (d) (d)
8
Sol : OD 2024
Sol : SQP 2016

Anti-derivative of x with respect to x is

3.
x 9. is equal to

(a) (b) (a) (b)

(c) (d) (c) (d)
Sol : OD 2023 Sol : Delhi 2010, OD 2008

4. If and f 1 1, then is 10. If

(a) x x (b) x x then the value of P is
(c) (d) x x (a) (b)
Sol : x OD 2023
(c) (d) 2
Sol : Comp 2007
5. equals
1 1
(a) (b) 11. is equal to
(c) (d)
Sol : OD 2020 (a) (b)
(c) (d) None of these
Sol : OD 2011
6. is equal to
(a) (b)
12. is equal to
(c) (d)
CHAPTER 7 Integrals Page 237

(a) (b) 19. If and , then

is equal to
(c) (d)
Sol : Comp 2017, Delhi 2008
(a) 2 (b) 2
(c) (d)
13. is equal to Sol : Comp 2018

(a) 1 (b) 2
(c) 2 (d) 0 20.
Sol : Foreign 2009
(a) (b)
(c) (d)
14. is equal to
Sol : OD 2014, Comp 2012

(a) a a (b) a a 2

(c) a a2 (d) a 21.

Sol : a SQP 2020

(a) (b)
1

1
Sol : Foreign 2007

22. If then

15. The value of is

e e (a) (b) 0
(a) (b)
e
(c) 1 (d) 1
(c) e (d) Sol : Comp 2016
e e
Sol : Foreign 2018, Delhi 2010

23.
16. The value of is
(a) 2 (b) 0 (a) 1 (b)
(c) 2 (d) 4
Sol : Delhi 2015 (c) (d) 0
Sol : OD 2012, Delhi 2010

17. is equal to
(a) 0 (b) 24.
4
(c) (d) (a) 0 (b) 1
2
Sol : OD 2008
(c) 2 (d)
Sol : Comp 2016

18. is equal to
(a) 0 (b) 2 25.
(c) 4 (d) 2
Sol : Foreign 2010, OD 2009 (a) (b) 2
Page 238 Integrals CHAPTER 7

Sol : SQP 2014

(c) (d)
Sol : OD 2008
33. The anti-derivative of is
(a) c (b) c
5
26.
(c) (d) None of these
6 6 Sol : Foreign 2017, Delhi 2009
(a) 5 5
(b)
6
6 6
(c) (d) 5 5
is
6 34.
Sol : Delhi 2007
(a) (b) x
(c) x (d) None
Sol : Comp 2008
27.

(a) 1 (b)
35. is equal to
(c) (d) None of these (a) x (b) x
Sol : Delhi 2010
(c) x (d) x
Sol : OD 2007

28.
2
(a) 0 (b) 2
(c) 1 (d) 4
Sol : Foreign 2017

29.
36. is equal to
(a) 1 (b) 1 (a) x x (b) x x
(c) 0 (d) None of these (c) x x x (d) x x x
Sol : Delhi 2012, Comp 2009 Sol : Foreign 2018, OD 2013

30. The integration of is 37. is equal to

(a) (b)
(a) x (b) x
(c) (d) None of these
Sol : Delhi 2008
(c) x (d) x
Sol : SQP 2020

31. Which one is equal to 3

?
38. The value of is equal to the value of:
(a) (b)
(a) (b)
(c) (d)
Sol : OD 2012
(c) (d)
Sol : OD 2012

32.

(b) x
5
(a) x5 is equal to
5 39.

(c) x (d) x (a) 0 (b)

CHAPTER 7 Integrals Page 239

(a) Both (A) and (R) are true and (R) is the correct
(c) 1 (d) None of these explanation of (A).
Sol : Delhi 2007
(b) Both (A) and (R) are true but (R) is not the
correct explanation of (A).
40. is equal to (c) (A) is true but (R) is false.
(d) (A) is false but (R) is true
(a) 0 (b) Sol : OD 2010
(c) 1 (d) None of these
Sol : Delhi 2018

45. Assertion (A) :

41. is equal to Reason (R) :

.
(a) (b) e
4 (a) Both (A) and (R) are true and (R) is the correct
(c) e (d) explanation of (A).
2
Sol : OD 2014, Foreign 2008 (b) Both (A) and (R) are true but (R) is not the
correct explanation of (A).
(c) (A) is true but (R) is false.
42. Assertion (A) :
(d) (A) is false but (R) is true
Sol : SQP 2020, Foreign 2017
Reason (R) : If we let then given function
becomes constant.
(a) Both (A) and (R) are true and (R) is the correct 46. Assertion (A) :
explanation of (A).
Reason (R) : Here x is odd function.
(b) Both (A) and (R) are true but (R) is not the
correct explanation of (A). (a) Both (A) and (R) are true and (R) is the correct
explanation of (A).
(c) (A) is true but (R) is false.
(b) Both (A) and (R) are true but (R) is not the
(d) (A) is false but (R) is true correct explanation of (A).
Sol : Comp 2017
(c) (A) is true but (R) is false.
(d) (A) is false but (R) is true
43. Assertion (A) : Sol : Delhi 2010

Reason (R) :
47. Assertion (A) :
(a) Both (A) and (R) are true and (R) is the correct
explanation of (A). Reason (R) : ,
(b) Both (A) and (R) are true but (R) is not the where is an even function.
correct explanation of (A). (a) Both (A) and (R) are true and (R) is the correct
(c) (A) is true but (R) is false. explanation of (A).
(d) (A) is false but (R) is true (b) Both (A) and (R) are true but (R) is not the
Sol : Delhi 2018 correct explanation of (A).
(c) (A) is true but (R) is false.
(d) (A) is false but (R) is true
44. Assertion (A) : Sol : Comp 2009

Reason (R) : Here is even function 48. Assertion (A) : Let then
if : .
Reason (R) : Integration is inverse process of
Page 240 Integrals CHAPTER 7

differentiation. 52. If n then

(a) Both (A) and (R) are true and (R) is the correct
explanation of (A). Assertion (A) :

(b) Both (A) and (R) are true but (R) is not the
correct explanation of (A). Reason (R) :
(c) (A) is true but (R) is false.
(d) (A) is false but (R) is true (a) Both (A) and (R) are true and (R) is the correct
Sol : Foreign 2009 explanation of (A).
(b) Both (A) and (R) are true but (R) is not the
correct explanation of (A).
49. Assertion (A) : (c) (A) is true but (R) is false.
1
(d) (A) is false but (R) is true
Reason (R) : Sol : Comp 2015, OD 2011
(a) Both (A) and (R) are true and (R) is the correct
explanation of (A).
(b) Both (A) and (R) are true but (R) is not the 53. Assertion (A) :
correct explanation of (A).
(c) (A) is true but (R) is false. Reason (R) : x is an odd function
(d) (A) is false but (R) is true (a) Both (A) and (R) are true and (R) is the correct
Sol : Delhi 2007 explanation of (A).
(b) Both (A) and (R) are true but (R) is not the
correct explanation of (A).
50. Assertion (A) : 1
2 1 (c) (A) is true but (R) is false.
(d) (A) is false but (R) is true
Reason (R) : Sol : SQP 2016

(a) Both (A) and (R) are true and (R) is the correct
explanation of (A).
(b) Both (A) and (R) are true but (R) is not the
correct explanation of (A).
(c) (A) is true but (R) is false.
(d) (A) is false but (R) is true
Sol : OD 2018

54. Assertion (A) :

51. Assertion (A) : 2

Reason (R) :
Reason (R) :
(a) Assertion is true, Reason is true; Reason is a
correct explanation for Assertion.
where
(b) Assertion is true, Reason is true; Reason is not a
correct explanation for Assertion.
(a) Both (A) and (R) are true and (R) is the correct (c) Assertion is true; Reason is false.
explanation of (A). (d) Assertion is false; Reason is true.
(b) Both (A) and (R) are true but (R) is not the Sol : OD 2017
correct explanation of (A).
(c) (A) is true but (R) is false.
(d) (A) is false but (R) is true 55. Assertion (A) :
Sol : OD 2012
Reason (R) : makes the integrand in I as a
rational function.
CHAPTER 7 Integrals Page 241

(a) Both (A) and (R) are true and (R) is the correct 67. Evaluate .
explanation of (A). Sol : OD 2011, Delhi 2008
(b) Both (A) and (R) are true but (R) is not the
correct explanation of (A).
(c) (A) is true but (R) is false. 68. Evaluate .
(d) (A) is false but (R) is true Sol : OD 2011

Sol : Foreign 2013

69. Evaluate .
Sol : Comp 2011
VERY SHORT ANSWER QUESTIONS

70. Write the value of .

Sol : Foreign 2011
56. Find :
Sol : OD 2024

71. Evaluate .
57. Find : Sol : Delhi 2010,
Sol : OD 2024

72. Evaluate .
58. Evaluate . Sol : Delhi 2010; OD 2010
Sol : OD 2023

73. Evaluate .
59. Find . Sol : Foreign 2010

Sol : Foreign 2014

74. Evaluate .
Sol : Foreign 2010, OD 2008
60. Find .
Sol : Foreign 2014

75. Evaluate 2 .
Sol : Foreign 2010
61. Evaluate .
Sol : Foreign 2014

76. Evaluate .
Sol : OD 2018
62. Evaluate .
Sol : Delhi 2012, Comp 2010

77. Find .
Sol : Delhi 2017
63. Evaluate .
Sol : Foreign 2012

78. Find : .
Sol : Foreign 2018
64. Write the value of .
Sol : Foreign 2011

79. Evaluate .
Sol : Delhi 2017
65. Write the value of .
Sol : Delhi 2011

80. Evaluate .
Sol : Foreign 2014, Delhi 2013
66. Write the value of .
Sol : Delhi 2011
Page 242 Integrals CHAPTER 7

81. Evaluate . 94. Find :

Sol : Foreign 2014
Sol : OD 2024

82. Evaluate .
Sol : Foreign 2014 95. Evaluate
Sol : OD 2023

83. Evaluate .
Sol : Comp 2014, OD 2011 x
96. Find
x
Sol : OD 2023

84. Evaluate .
Sol : Comp 2014
97. Evaluate
Sol : OD 2023

85. Evaluate .
Sol : Delhi 2012
98. Find the value of .
Sol : OD 2020

86. Evaluate .
Sol : Comp 2011
99. Evaluate .
Sol : OD 2020

87. Evaluate .
Sol : OD 2012

88. Evaluate .
Sol : Foreign 2011, Delhi 2009

89. Evaluate .
Sol : Delhi 2014 100. Find the value of
Sol : OD 2020

90. Evaluate .
Sol : OD 2014 101. Find .
Sol : OD 2017, Delhi 2010

91. Write the value of .

Sol : Comp 2012 102. Evaluate .
Sol : Delhi 2014C; Foreign 2014

92. Evaluate .
Sol : OD 2011C, 2008 103. Write the anti-derivative of x .
Sol : x Delhi 2014

SHORT ANSWER QUESTIONS 104. Given, . Write

satisfying above.
Sol : OD 2012; Foerign 2011

93. Evaluate :
Sol : OD 2024 105. Write the value of .
Sol : Comp 2012
CHAPTER 7 Integrals Page 243

106. Write the value of . 120. Find .

Sol : Delhi 2012C, 2011
Sol : OD 2019, Delhi 2017

121. Find .
107. Evaluate .
Sol : OD 2019
Sol : OD 2011

122. Find .
108. Evaluate . Sol : OD 2019
Sol : OD 2011 , Foreign 2007

123. Find .
109. Evaluate . Sol : OD 2017

Sol : Foreign 2011; Delhi 2009

124. Find .
2 2
Sol : 3 3
Delhi 2016, OD 2012
110. Evaluate 2 2 .
Sol : Foreign 2011

125. Evaluate .
Sol : Foreign 2016; Delhi 2012

111. Evaluate .
Sol : Foreign 2011
126. Evaluate .
Sol : Foreign 2015; Delhi 2013

112. Evaluate .
Sol : Comp 2011
127. Find .
Sol : Delhi 2014

113. Evaluate .
Sol : OD 2011C, 2009, 2009
128. Find .
Sol : Comp 2014

114. Find : .
Sol : Delhi 2019
129. Evaluate .
Sol : OD 2014C; Foreign 2014

115. Find : , x .
Sol : Delhi 2019, Comp 2010
130. Evaluate .
Sol : Comp 2014, Delhi 2010

116. Find : .
Sol : Delhi 2019
131. Evaluate .
Sol : OD 2013

117. Find the value of .

Sol : Delhi 2019
132. Evaluate .
Sol : OD 2013

118. Find the values of .

Sol : Delhi 2019
133. Evaluate .
Sol : OD 2013

119. Find .
Sol : OD 2019
Page 244 Integrals CHAPTER 7

134. Evaluate . 148. Evaluate .

Sol : OD 2013
Sol : Delhi 2019

135. Evaluate .
Sol : Delhi 2012, Comp 2009 149. Evaluate .
Sol : Delhi 2016; OD 2010

136. Evaluate . 150. Evaluate .

Sol : Comp 2013 Sol : Comp 2015, OD 2010, Comp 2013

137. Evaluate . 151. Evaluate .

Sol : Comp 2012
Sol : OD 2016

138. Evaluate . 152. Evaluate .

Sol : Comp 2011, Foreign 2008 Sol : Comp 2014

139. Evaluate . 153. Evaluate .

Sol : Delhi 2010

Sol : Delhi 2013

140. If , then find the value of a .

154. Evaluate .
Sol : OD 2013
Sol : OD 2014, Delhi 2010

155. Evaluate .
141. If , then write the value of . Sol : Delhi 2013
Sol : OD 2014

156. Evaluate .
142. Evaluate . Sol : Delhi 2013

Sol : OD 2014
157. Evaluate .
Sol : Delhi 2011C, 2008, OD 2009

143. Evaluate .
Sol : Delhi 2014
158. Evaluate .
Sol : OD 2011

144. Evaluate .
Sol : Comp 2014, OD 2009
159. Evaluate .
Sol : Comp 2013

145. Evaluate .
Sol : Comp 2010
160. Evaluate .
Sol : Comp 2011, OD 2007

146. Write the value of the following integral .

Sol : OD 2010
161. Evaluate .
Sol : Comp 2011

147. Evaluate .
Sol : Delhi 2019
CHAPTER 7 Integrals Page 245

162. Evaluate . 173. Find d .

Sol : Delhi 2010 Sol : OD 2017, Comp 2015

163. Evaluate . 174. Find d .

Sol : OD 2017, Comp 2011
Sol : Delhi 2016

LONG ANSWER QUESTIONS 175. Find .

Sol : OD 2016, Delhi 2015, Comp 2014

176. Evaluate .
164. Find Sol : OD 2016, Delhi 2015, Comp 2009
Sol : OD 2020

165. Find : . 177. Find .

Sol : Delhi 2019 Sol : OD 2016

178. Find .
Sol : Foreign 2016

179. Find .
Sol : Foreign 2016, OD 2007

166. Find the value of .

Sol : Delhi 2019 180. Find .
Sol : Delhi 2015

Find .
Integrate w.r.t. x , x x
167.
Sol : OD 2019, Delhi 2008 181. .
Sol : x Delhi 2015

168. Find . 182. Evaluate .

Sol : OD 2019
Sol : OD 2015

169. Find .
Sol : OD 2018, Foreign 2007 183. Find .
Sol : OD 2015, Delhi 2010

170. Find .
Sol : Comp 2018 184. .
Sol : Foreign 2015

171. Find .
Sol : Delhi 2017 185. Evaluate .

Sol : Foreign 2015; Delhi 2013

172. Find .
Sol : Delhi 2017
186. Evaluate .
Sol : Delhi 2014
Page 246 Integrals CHAPTER 7

187. Evaluate . 201. Evaluate .

Sol : OD 2014 Sol : Comp 2010

188. Evaluate . 202. Evaluate .

Sol : Foreign 2014 Sol : Comp 2010

189. Find . 203. Evaluate .

Sol : Delhi 2014C; Delhi 2013 Sol : OD 2010

190. Evaluate . 204. Evaluate .

Sol : Comp 2013 Sol : OD 2014, Delhi 2010

191. Evaluate . 205. Evaluate .

Sol : Delhi 2013 Sol : OD 2014, Delhi 2010

192. Evaluate . 206. Evaluate .

Sol : Delhi 2013, SQP 2012
Sol : Delhi 2014, Comp 2010

193. Evaluate .
Sol : OD 2013

194. Evaluate .
Sol : Comp 2013, OD 2011

207. Find .
Sol : Delhi 2014
195. Evaluate .
Sol : Delhi 2012

208. Find ,x .
196. Evaluate . Sol : OD 2014

Sol : Comp 2012

209. Find .
Sol : Comp 2014, OD 2010

197. Evaluate .
Sol : Foreign 2011

210. Find .
198. Evaluate .
Sol : Foreign 2011, Delhi 2008 Sol : Foreign 2014

199. Evaluate . 211. Evaluate .

Sol : Delhi 2012

Sol : Delhi 2011

212. Evaluate .
Sol : OD 2011, Foreign 2009
200. Evaluate .
Sol : Delhi 2011; OD 2010
CHAPTER 7 Integrals Page 247

213. Prove that , hence evaluate 226. Evaluate .

Sol : OD 2011
.
Sol : Delhi 2019

227. Evaluate .
Sol : Foreign 2011
214. Prove that . and hence
evaluate .
Sol : OD 2019, Delhi 2007 228. Evaluate .
Sol : OD 2011

215. Evaluate .
Sol : Delhi 2017; OD 2013, 2012, 2011C, 2008 229. Evaluate .
Sol : Delhi 2010

216. Evaluate . 230. Evaluate .

Sol : OD 2017 Sol : OD 2018

217. Evaluate . 231. Evaluate .

Sol : Delhi 2016 Sol : Foreign 2014; Delhi 2014, 2011

218. Evaluate . 232. Evaluate .

Sol : Foreign 2016, OD 2014 Sol : Foreign 2014; OD 2009

219. Evaluate . 233. Evaluate .

Sol : Delhi 2015 Sol : Foreign 2014; Delhi 2014C, 2010, 2008; OD 2008

220. Find . 234. Evaluate .

Sol : OD 2015 Sol : Delhi 2014

221. Evaluate . 235. Prove that .

Sol : Foreign 2015 Sol : Delhi 2012

222. Evaluate . 236. Evaluate .

Sol : OD 2014C; Delhi 2011 Sol : Foreign 2012

223. Prove that . 237. Evaluate .

Sol : Foreign 2014 Sol : Delhi 2011

Evaluate .
224.
Sol : Foreign 2011 CASE BASED QUESTIONS

225. Evaluate . 238. Commodity prices are primarily determined by the

Sol : OD 2011, Comp 2008
forces of supply and demand in the market. For
example, if the supply of oil increases, the price of
one barrel decreases. Conversely, if demand for oil
Page 248 Integrals CHAPTER 7

increases (which often happens during the summer), any time t . Initially t 0 50 g of the first substance
the price rises. Gasoline and natural gas fall into the was present; 1 hr later, only 10 g of it remained.
energy commodities category. (i) Find an expression that gives the amount of the
first substance present at any time t .
(ii) What is the amount present after 2 hr?
Sol :

240. Chemical reaction, a process in which one or more

substances, the reactants, are converted to one or more
different substances, the products. Substances are
either chemical elements or compounds. A chemical
reaction rearranges the constituent atoms of the
reactants to create different substances as products.

The price p (dollars) of each unit of a particular

commodity is estimated to be changing at the rate
x
x
where x (hundred) units is the consumer demand (the
number of units purchased at that price). Suppose 400
units x 4 are demanded when the price is $30 per
unit.
(i) Find the demand function .
(ii) At what price will 300 units be demanded? At
what price will no units be demanded? In a certain chemical reaction, a substance is converted
(iii) How many units are demanded at a price of $20 into another substance at a rate proportional to the
per unit? square of the amount of the first substance present at
Sol :
any time t . Initially t 0 50 g of the first substance
was present; 1 hr later, only 10 g of it remained.
239. Chemical reaction, a process in which one or more (i) Find an expression that gives the amount of the
substances, the reactants, are converted to one or more first substance present at any time t .
different substances, the products. Substances are (ii) What is the amount present after 2 hr?
Sol :
either chemical elements or compounds. A chemical
reaction rearranges the constituent atoms of the
reactants to create different substances as products. 241. Bata India is the largest retailer and leading
manufacturer of footwear in India and is a part of the
Bata Shoe Organization. Incorporated as Bata Shoe
Company Private Limited in 1931, the company was
set up initially as a small operation in Konnagar (near
Calcutta) in 1932. In January 1934,

In a certain chemical reaction, a substance is converted

into another substance at a rate proportional to the
square of the amount of the first substance present at
CHAPTER 7 Integrals Page 249

(i) How many items can Bob memorize during the

first 10 minutes?
(ii) How many additional items can he memorize
during the next 10 minutes (from time t to
t )?
Sol :

243. Different types of drugs affect your body in different

ways, and the effects associated with drugs can vary
from person to person. How a drug effects an individual
is dependent on a variety of factors including body
size, general health, the amount and strength of the
The manager of BATA show room at Jaipur determines drug, and whether any other drugs are in the system
that the price p (dollars) for each pair of a popular at the same time. It is important to remember that
brand of sports sneakers is changing at the rate of illegal drugs are not controlled substances, and
x therefore the quality and strength may differ from one
x batch to another.
when x (hundred) pairs are demanded by consumers.
When the price is $75 per pair, 400 pairs x 4 are
demanded by consumers.
(i) Find the demand (price) function .
(ii) At what price will 500 pairs of sneakers be
demanded? At what price will no sneakers be
demanded?
(iii) How many pairs will be demanded at a price of
$90 per pair?
Sol :

The concentration in milligrams per cubic

242. Bob is taking a learning test in which the time he centimeter of a drug in a patient’s
takes to memorize items from a given list is recorded. bloodstream is immediately after an
Let be the number of items he can memorize in injection and t minutes later is decreasing at the rate
t minutes. His learning rate is found to be
per minute
t t
A new injection is given when the concentration drops
below .
(i) Find an expression for .
(ii) What is the concentration after 1 hour? After 3
hours?
Sol :

244. In mathematics, a continuous function is a function

such that a continuous variation of the argument
induces a continuous variation of the value of the
function. This means that there are no abrupt changes
Page 250 Integrals CHAPTER 7

in value, known as discontinuities.

Let be a continuous function defined on ,

then

On the basis of above information, answer the

following questions.
(i) Evaluate

(ii) Evaluate

iii) Evaluate ]

(iv) Evaluate

Sol :

***********