Page 1 :
Chapter 7 Integrals, EXERCISE 7.1, , Question 1:, Find an anti-derivative (or integral) of the following functions by the method of inspection,, ., , Solution:, , Thus, the anti-derivative of sin 2x is, Question 2:, Find an anti-derivative (or integral) of the following functions by the method of inspection,, ., , Solution:, , Thus, the anti-derivative of, , is, , Question 3:, Find an anti-derivative (or integral) of the following functions by the method of inspection,, .
Page 2 :
Solution:, , Thus, the anti-derivative of, , is, , ., , Question 4:, Find an anti-derivative (or integral) of the following functions by the method of inspection,, ., Solution:, , Thus, the anti-derivative, , of is, , Question 5:, Find an anti-derivative (or integral) of the following functions by the method of inspection,, , Solution:, , Thus, the anti-derivative of, , is, , Find the following integrals in Exercises 6 to 20:, Question 6:
Page 3 :
Solution:, , Question 7:, , Solution:, , Question 8:, , Solution:, , Question 9:
Page 4 :
Solution:, , Question 10:, , Solution:, , Question 11:, , Solution:, , Question 12:
Page 5 :
Solution:, , Question 13:, , Solution:, , Question 14:, , Solution:
Page 6 :
Question 15:, , Solution:, , Question 16:, , Solution:, , Question 17:, , Solution:
Page 7 :
Question 18:, , Solution:, , Question 19:, , Solution:
Page 8 :
Question 20:, , Solution:, , Choose the correct answer in Exercises 21 and 22, , Question 21:, The anti-derivative of, , Solution:, , Thus, the correct option is C., , equals
Page 9 :
Question 22:, If, , such that, , Solution:, Given,, Anti-derivative of, Therefore,, , Also,, , Thus, the correct option is A., , , then, , is
Page 10 :
EXERCISE 7.2, , Integrate the functions in Exercises 1 to 37:, Question 1:, , Solution:, Put, Therefore,, , Question 2:, , Solution:, Put, Therefore,, , Question 3:, , Solution:, , Put
Page 11 :
Therefore,, , Question 4:, , Solution:, Put, Therefore,, , Question 5:, , Solution:, , Put, Therefore,, , Question 6:, Solution:, Put, Therefore,
Page 12 :
Question 7:, , Solution:, , Question 8:
Page 13 :
Solution:, , Question 9:, , Solution:, , Question 10:
Page 14 :
Solution:, , Question 11:, , Solution:
Page 15 :
Question 12:, , Solution:, , Question 13:, , Solution:
Page 16 :
Question 14:, , Solution:, , Question 15:, , Solution:, Put,, , Question 16:, , Solution:, Put,
Page 17 :
Question 17:, , Solution:, Put,, , Question 18:, , Solution:, Put,, , Question 19:, , Solution:, , Dividing Nr and Dr by, , , we get
Page 18 :
Let, , Question 20:, , Solution:, Put,, , Question 21:, , Solution:, Put,
Page 19 :
Question 22:, , Solution:, Put,, , Question 23:, , Solution:, Put,
Page 20 :
Question 24:, , Solution:, , Question 25:, , Solution:
Page 21 :
Question 26:, , Solution:, Let, , Question 27:, , Solution:, Put,, So,, , Question 28:, , Solution:, Put,
Page 22 :
Question 29:, , Solution:, Let, , Question 30:, , Solution:, Put,
Page 23 :
Question 31:, , Solution:, Put,, , Question 32:, , Solution:, Let I
Page 24 :
Question 33:, , Solution:, Put, I, , Question 34:, , Solution:, , Question 35:
Page 25 :
Solution:, Put,, , Question 36:, , Solution:, , Question 37:, , Solution:, Put,, , Let
Page 26 :
From (1), we get, , Choose the correct answer in Exercises 38 and 39., Question 38:, equals, , Solution:, Put,, , Thus, the correct option is D., Question 39:, equals, , Solution:, Put,
Page 27 :
Thus, the correct option is B.
Page 28 :
EXERCISE 7.3, Find the integrals of the functions in Exercises 1 to 22:, Question 1:, , Solution:, , Question 2:, Solution:, Using,, , Question 3:
Page 29 :
Solution:, Using,, , Question 4:, , Solution:, Put, I
Page 30 :
Question 5:, , Solution:, Let I, , Let, , Question 6:, Solution:, Using,
Page 31 :
Question 7:, Solution:, Using,, , Question 8:, , Solution:
Page 32 :
Question 9:, , Solution:, , Question 10:
Page 33 :
Solution:, , Question 11:
Page 34 :
Solution:, , Question 12:, , Solution:
Page 35 :
Question 13:, , Solution:, , Question 14:, , Solution:, , Let
Page 36 :
Question 15:, , Solution:
Page 37 :
Question 16:, , Solution:, , Consider, Let, , From equation (1), we get, , Question 17:, , Solution:
Page 38 :
Question 18:, , Solution:, , Question 19:, , Solution:, , Let
Page 39 :
Question 20:, , Solution:, , Let, , Question 21:, Solution:, Let, Then,
Page 40 :
Let, , We know that,, , Substituting in equation (1), we get
Page 41 :
Question 22:, , Solution:, , Choose the correct answer in Exercises 23 and 24., Question 23:, is equal to, , Solution:, , Thus, the correct option is A.
Page 42 :
Question 24:, equals, , Solution:, , Put,, , Thus, the correct answer is B.
Page 43 :
EXERCISE 7.4, Integrate the functions in Exercises 1 to 23, Question 1:, , Solution:, Put,, , Question 2:, , Solution:, Put,, , Question 3:, , Solution:, Put,
Page 44 :
Question 4:, , Solution:, Put,, , Question 5:, , Solution:, Let
Page 45 :
Question 6:, , Solution:, Put,, , Question 7:, , Solution:, , For, , let
Page 46 :
From (1), we get, , Question 8:, , Solution:, Put,, , Question 9:, , Solution:, Put,
Page 47 :
Question 10:, , Solution:, , Let, , Question 11:
Page 48 :
Solution:, , Question 12:, , Solution:, , can be written as, Thus,
Page 49 :
Question 13:, , Solution:, , can be written as, , Thus,, , Let, , Question 14:, , Solution:
Page 50 :
Thus,, , Question 15:, , Solution:, Thus,
Page 51 :
Let, , Question 16:, , Solution:, , Equating the coefficients of x and constant term on both sides, we get
Page 52 :
Question 17:, , Solution:, , Equating the coefficients of x and constant term on both sides, we get, , From (1) we get, , Then,, From equation (2) we get, , Question 18:
Page 53 :
Solution:, Let, Equating the coefficients of x and constant term on both sides, we get, , Let, , can be written as, Thus,
Page 54 :
Substituting equations (2) and (3) in equation (1), we get, , Question 19:, , Solution:
Page 55 :
Put,, Equating the coefficients of x and constant term, we get, , Let, , Then,, , and
Page 56 :
Thus,, , Substituting equations (2) and (3) in (1), we get, , Question 20:, , Solution:, Consider,, Equating the coefficients of x and constant term on both sides, we get, , Let
Page 57 :
Then,, , Using equations (2) and (3) in (1), we get, , Question 21:, , Solution:
Page 58 :
Let, , Put,, , Using equations (2) and (3) in (1), we get, , Question 22:, , Solution:, Let, Equating the coefficients of x and constant term on both sides, we get
Page 59 :
Let, , Then,, Put,, , Substituting (2) and (3) in (1), we get, , Question 23:
Page 60 :
Solution:, Let, Equating the coefficients of x and constant term, we get, , Let, , Then,, , Put,, , Using equations (2) and (3) in (1), we get
Page 61 :
Choose the correct answer in Exercises 24 and 25., Question 24:, equals, , Solution:, , Hence, the correct option is B., Question 25:, equals
Page 62 :
Solution:, , Hence, the correct option is B.
Page 63 :
EXERCISE 7.5, , Integrate the rational functions in Exercises 1 to 21., Question 1:, , Solution:, Let, Equating the coefficients of x and constant term, we get, On solving, we get, , Question 2:, , Solution:, Let, Equating the coefficients of x and constants term, we get, On solving, we get
Page 64 :
Question 3:, , Solution:, Let, Equating the coefficients of, , Solving these equations, we get, , Question 4:, , Solution:, Let, , and constant terms, we get
Page 65 :
and constant terms, we get, , Equating the coefficients of, , Solving these equations, we get, , Question 5:, , Solution:, , Equating the coefficients of x and constant terms, we get, Solving these equations, we get, , Question 6:, , Solution:, It can be seen that the given integrand is not a proper fraction., Therefore, on dividing, , by, , , we get
Page 66 :
Let, Equating the coefficients of x and constant term, we get, , Solving these equations, we get, , Substituting in equation (1), we get, , Question 7:, , Solution:, Let, , Equating the coefficients of, , and constant term, we get, , On solving these equations, we get, From equation (1), we get
Page 67 :
Consider, , , let, , Question 8:, , Solution:, Let, Equating the coefficients of, , and constant term, we get, , On solving these equations, we get
Page 68 :
Question 9:, , Solution:, , Let, , Equating the coefficients of, , and constant term, we get, , On solving these equations, we get, , Question 10:, , Solution:, , Let
Page 69 :
Equating the coefficients of, , and constant term, we get, , On solving, we get, , Question 11:, , Solution:, , Let, Equating the coefficients of, , On solving, we get, , and constant term, we get
Page 70 :
Question 12:, , Solution:, On dividing, , by, , , we get, , Let, Equating the coefficients of, On solving, we get, , Question 13:, , and constant term, we get
Page 71 :
Solution:, Let, , Equating the coefficients of, , and constant term, we get, , On solving these equations, we get, , Question 14:, , Solution:, Let, Equating the coefficient of x and constant term, we get
Page 72 :
Question 15:, , Solution:, , Let, , Equating the coefficients of, , On solving, we get, , and constant term, we get
Page 73 :
Question 16:, , [Hint: multiply numerator and denominator by, , and put, , Solution:, , Multiplying numerator and denominator by, , , we get, , Equating the coefficients of t and constant term, we get, , ]
Page 74 :
Question 17:, [Hint: Put, , ], , Solution:, Put,, , Let, Equating the coefficients of t and constant, we get, On solving, we get
Page 75 :
Question 18:, , Solution:, , Let, , Equating the coefficients of, , On solving these equations, we get, , Question 19:, , and constant term, we get
Page 76 :
Solution:, , Put,, , Equating the coefficients of t and constant, we get, On solving, we get, , Question 20:, , Solution:, , Multiplying Nr and Dr by, , , we get
Page 77 :
Equating the coefficients of t and constant, we get, , Question 21:, , [Hint: Put, , ], , Solution:, Put, , Equating the coefficients of t and constant, we get
Page 78 :
Question 22:, equals, , Solution:, Let, Equating the coefficients of x and constant, we get, , Thus, the correct option is B.
Page 79 :
Question 23:, , equals, , Solution:, Let, Equating the coefficients of, , Thus, the correct option is A., , and constant terms, we get
Page 80 :
EXERCISE 7.6, , Integrate the functions in Exercises 1 to 22., Question 1:, Solution:, Let, Taking, , and, , and integrating by parts,, , Question 2:, Solution:, Let, Taking, , and, , and integrating by parts,, , and, , and integrating by parts, we get, , Question 3:, , Solution:, Let, Taking
Page 81 :
Again using integration by parts, we get, , Question 4:, , Solution:, Let, Taking, , and, , and integrating by parts, we get, , Question 5:, , Solution:, Let, Taking, , and, , and integrating by parts, we get
Page 82 :
Question 6:, , Solution:, Let, Taking, , and, , and integrating by parts, we get, , Question 7:, , Solution:, Let, Taking, , and, , and integrating by parts, we get
Page 83 :
Question 8:, , Solution:, Let, Taking, , and, , and integrating by parts, we get
Page 84 :
Question 9:, , Solution:, Let, Taking, , and, , and integrating by parts, we get
Page 85 :
Question 10:, , Solution:, Let, , Taking, , and, , and integrating by parts, we get
Page 86 :
Question 11:, , Solution:, Let, , Taking, , Question 12:, , Solution:, Let, , and, , and integrating by parts, we get
Page 87 :
Taking, , and, , and integrating by parts, we get, , Question 13:, , Solution:, Let, Taking, , and, , and integrating by parts, we get, , Question 14:, , Solution:, Let, Taking, , and, , and integrating by parts, we get, , Again, using integration by parts, we get
Page 88 :
Question 15:, , Solution:, Let, Let, , Where,, , Taking, , and, , Taking, , and, , Using equations, , and integrating by parts, we get, , and, , and integrating by parts,, , in, , ,
Page 89 :
Question 16:, , Solution:, Let, Let, , Question 17:, , Solution:
Page 90 :
Question 18:, , Solution:, , Let, It is known that,, From equation, , Question 19:, , Solution:, Let, Here,, It is known that,, , , we get
Page 91 :
Question 20:, , Solution:, , Let, It is known that,, , Question 21:, , Solution:, Let, Taking, , and, , and integrating by parts, we get, , Again, using integration by parts, we get
Page 92 :
Question 22:, , Solution:, Let
Page 93 :
Using integration by parts, we get, , Question 23:, equals, A., , B., C., D., , Solution:, Let, Also, let, , Thus, the correct option is A.
Page 94 :
Question 24:, equals, A., B., C., D., , Solution:, Consider,, Let, , It is known that,, , Thus, the correct option is B.
Page 95 :
EXERCISE 7.7, Integrate the functions in Exercises 1 to 9., Question 1:, , Solution:, Let, , Since,, , Question 2:, , Solution:, , Since,
Page 96 :
Question 3:, , Solution:, , Question 4:, , Solution:, Consider,
Page 97 :
Question 5:, , Solution:, , Question 6:, , Solution:, , Question 7:, , Solution:, Put,
Page 98 :
Question 8:, , Solution:, , Question 9:, , Solution:
Page 99 :
Question 10:, , is equal to, , Solution:, , Thus, the correct option is A., Question 11:, , is equal to, , Solution:, , Thus, the correct option is D.
Page 100 :
EXERCISE 7.8, , Evaluate the following definite integrals as limit of sums., Question 1:, , Solution:, Since,, Here,, , where
Page 101 :
Question 2:, , Solution:, Let, , Since,, , , where, , Here,, , Question 3:, , Solution:, Since,, , where, Here,
Page 102 :
Question 4:, , Solution:, Let, , Let, , Since,, For,, , where, , , where
Page 103 :
For
Page 104 :
From equations (2) and (3), we get, , Question 5:, , Solution:, Let, , Since,, , Here,, , , where
Page 105 :
Question 6:, , Solution:, Since,, , , where, , Here,
Page 107 :
EXERCISE 7.9, , Evaluate the definite integrals in Exercises 1 to 20., Question 1:, , Solution:, Let, , Using second fundamental theorem of calculus, we get, , Question 2:, , Solution:, Let, Using second fundamental theorem of calculus, we get, , Question 3:, , Solution:, Let
Page 108 :
Using second fundamental theorem of calculus, we get, , Question 4:, , Solution:, , Using second fundamental theorem of calculus, we get
Page 109 :
Question 5:, , Solution:, Let, , Using second fundamental theorem of calculus, we get, , Question 6:, , Solution:, Let, Using second fundamental theorem of calculus, we get, , Question 7:, , Solution:, Let, Using second fundamental theorem of calculus, we get
Page 110 :
Question 8:, , Solution:, Let, Using second fundamental theorem of calculus, we get, , Question 9:, , Solution:, Let, , Using second fundamental theorem of calculus, we get
Page 111 :
Question 10:, , Solution:, Let, Using second fundamental theorem of calculus, we get, , Question 11:, , Solution:, , Using second fundamental theorem of calculus, we get
Page 112 :
Question 12:, , Solution:, Let, , Using second fundamental theorem of calculus, we get, , Question 13:, , Solution:, Let, Using second fundamental theorem of calculus, we get, , Question 14:, , Solution:, Let
Page 113 :
Using second fundamental theorem of calculus, we get, , Question 15:, , Solution:, Let, Put,, As, , and as, , Using second fundamental theorem of calculus, we get
Page 114 :
Question 16:, , Solution:, Let, Dividing, , , we get, , Let, Equating the coefficients of x and constant term, we get, Let, , Substituting the value, , in (1), we get
Page 115 :
Question 17:, , Solution:, Let, , Using second fundamental theorem of calculus, we get, , Question 18:, , Solution:, Let, , Using second fundamental theorem of calculus, we get
Page 116 :
Question 19:, , Solution:, Let, , Using second fundamental theorem of calculus, we get, , Question 20:, , Solution:, Let
Page 117 :
Using second fundamental theorem of calculus, we get, , Question 21:, , Solution:, Using second fundamental theorem of calculus, we get, , Thus, the correct option is D.
Page 118 :
Question 22:, , Solution:, , Put, , Using second fundamental theorem of calculus, we get, , Thus, the correct option is C.
Page 119 :
EXERCISE 7.10, , Evaluate the integrals in Exercises 1 to 8 using substitution., Question 1:, , Solution:, Put,, When,, , Question 2:, , Solution:, Consider,, Let, , When, , and when
Page 120 :
Question 3:, , Solution:, Consider,, Let, , When, , Taking, , and, , and integrating by parts, we get
Page 121 :
Question 4:, , Solution:, Put,
Page 122 :
Question 5:, , Solution:, , Put,, , When, , Question 6:, Solution:, , Let
Page 123 :
Question 7:, , Solution:, , Put,, , When
Page 124 :
Question 8:, , Solution:, , Put,, When, , Let, , Then,, , Question 9:, The value of the integral, A. 6, B. 0, C. 3, D. 4, , is
Page 125 :
Solution:, Consider,, Let, and when, , When, , Put, , and when, , When, , Thus, the correct option is A., Question 10:, If, , , then, , is
Page 126 :
Solution:, Using integration by parts, we get, , Thus, the correct option is B.
Page 127 :
EXERCISE 7.11, , By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19., Question 1:, , Solution:, , Adding (1) and (2), we get, , Question 2:, , Solution:, , Consider,
Page 128 :
Adding (1) and (2), we get, , Question 3:, , Solution:, , Let, , Adding (1) and (2), we get
Page 129 :
Question 4:, , Solution:, Consider,, , Adding (1) and (2), we get, , Question 5:, , Solution:, Let, , As,, , on, , and, , on
Page 130 :
Question 6:, , Solution:, Consider,, , As, , Question 7:, , Solution:, , Consider,
Page 131 :
Question 8:, , Solution:, , Let
Page 132 :
Question 9:, , Solution:, , Consider,, , Question 10:, , Solution:, Consider,, , Since,, Adding (1) and (2), we get
Page 133 :
Question 11:, , Solution:, Let, , , therefore, , As, , If, , is an even function, then, , is an even function.
Page 134 :
Question 12:, , Solution:, Let, , Adding (1) and (2), we get, , Question 13:, , Solution:, Let, , As, , , thus, , is an odd function, then, , is an odd function.
Page 135 :
Question 14:, , Solution:, Let, We know that,, , , if, , if, , Question 15:, , Solution:, Consider,, , Adding (1) and (2), we get
Page 136 :
Question 16:, , Solution:, Consider,, , Adding (1) and (2), we get, , Adding (4) and (5), we get
Page 137 :
Question 17:, , Solution:, Let, , We know that,, , Adding (1) and (2), we get, , …(1)
Page 138 :
Question 18:, , Solution:, Since,, , Question 19:, Show, , that, , Solution:, Let, , Adding (1) and (2), we get, , if, , f, , and, , g, , are, , defined, , as
Page 139 :
Question 20:, The value of, A. 0, B. 2, C., D. 1, , Solution:, Consider,, , For, , If, And, , an even function, then, , is an odd function, then, , Thus, the correct is option C., , is
Page 140 :
Question 21:, The value of, A. 2, , B., C. 0, D. -2, , Solution:, Let, , Adding (1) and (2), we get, , Thus, the correct option is C., , is
Page 141 :
MISCELLANEOUS EXERCISE, , Integrate the functions in Exercises 1 to 24., Question 1:, , Solution:, , Let, , Equating the coefficients of, , and constant terms, we get, , On solving these equations, we get, , From equation (1), we get
Page 142 :
Question 2:, , Solution:, , Question 3:, , Solution:, , Let
Page 143 :
Question 4:, , Solution:, , Multiplying and dividing by, , Let, , , we get
Page 144 :
Question 5:, , Solution:, , Let
Page 145 :
Question 6:, , Solution:, Consider,, , Equating the coefficients of, , and constant term, we get, , On solving these equations, we get, , From equation (1), we get
Page 146 :
Question 7:, , Solution:, , Put,
Page 147 :
Question 8:, , Solution:, , Question 9:, , Solution:, , Put,
Page 148 :
Question 10:, , Solution:, , Question 11:, , Solution:, , Multiplying and dividing by, , , we get
Page 149 :
Question 12:, , Solution:, , Put,
Page 150 :
Question 13:, , Solution:, , Put, , Question 14:, , Solution:, , Equating the coefficients of, , On solving these equations, we get, , and constant term, we get
Page 151 :
From equation (1), we get, , Question 15:, , Solution:, Let, , Question 16:, , Solution:, , Let
Page 152 :
Question 17:, , Solution:, , Put,, , Question 18:, , Solution:, , Put,
Page 153 :
Question 19:, , Solution:, Let, , As we know that,, , Let, Also, let
Page 154 :
From equation (1), we get, , Question 20:, , Solution:, , Put,
Page 156 :
Question 21:, , Solution:, , Let, , Question 22:, , Solution:, Let, , Equating the coefficients of, , and constant term, we get, , On solving these equations, we get, , From equation (1), we get
Page 157 :
Question 23:, , Solution:, , Let
Page 158 :
Question 24:, , Solution:, , Let, , Using integration by parts, we get
Page 159 :
Question 25:, , Solution:, , Let, , Question 26:, , Solution:, Let
Page 160 :
Put,, When, , Question 27:, , Solution:, Consider,, , and when
Page 161 :
Consider,, Put,, When, , and when, , Therefore, from (1), we get
Page 162 :
Question 28:, , Solution:, Consider,, , Let, , When, , and when, , As, , , therefore,, , We know that if, , is an even function, then, , is an even function
Page 163 :
Question 29:, , Solution:, Consider,, , Question 30:, , Solution:, Consider,, Put,, , When, , and when
Page 164 :
Question 31:, , Solution:, Consider,, Put,, When, , Consider, , and when
Page 165 :
Question 32:, , Solution:, Let, , Adding (1) and (2), we get
Page 166 :
Question 33:, , Solution:, Consider,, , Where,
Page 167 :
From equations (1), (2), (3) and (4), we get, , Question 34:, , Solution:, Consider,, , Let,
Page 168 :
Equating the coefficients of, , and constant terms, we get, , On solving these equations, we get, , Hence proved., Question 35:, , Solution:, Let, Using integration by parts, we get, , Hence proved.
Page 169 :
Question 36:, , Solution:, Consider,, , Let, , is an odd function., We know that if, Hence proved., Question 37:, , Solution:, Consider,, , Hence proved., , is an odd function, then
Page 170 :
Question 38:, , Solution:, Consider,, , Hence proved., Question 39:, , Solution:, Let, Using integration by parts, we get, , Put,, When, , Hence proved., , and when
Page 171 :
Question 40:, , as a limit of a sum., , Evaluate, , Solution:, Let, We know that,, , Where,, , and, , Here,, , Question 41:, , is equal to
Page 172 :
Solution:, , Consider,, , Put,, , Thus, the correct option is A., Question 42:, , is, , Equals to, Solution:, , Consider,, , Let
Page 173 :
Thus, the correct option is B., , Question 43:, , If, , is equal to, , , then, , Solution:, Consider,, , Thus, the correct option is D., Question 44:, The value of, A. 1, , is
Page 174 :
B. 0, C. -1, D., , Solution:, Consider,, , Adding (1) and (2), we get, , Thus, the correct option is B.
