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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,, .
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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:
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Solution:, , Question 7:, , Solution:, , Question 8:, , Solution:, , Question 9:
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Solution:, , Question 10:, , Solution:, , Question 11:, , Solution:, , Question 12:
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Solution:, , Question 13:, , Solution:, , Question 14:, , Solution:
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Question 15:, , Solution:, , Question 16:, , Solution:, , Question 17:, , Solution:
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Question 18:, , Solution:, , Question 19:, , Solution:
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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
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Question 22:, If, , such that, , Solution:, Given,, Anti-derivative of, Therefore,, , Also,, , Thus, the correct option is A., , , then, , is
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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
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Therefore,, , Question 4:, , Solution:, Put, Therefore,, , Question 5:, , Solution:, , Put, Therefore,, , Question 6:, Solution:, Put, Therefore,
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Question 7:, , Solution:, , Question 8:
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Solution:, , Question 9:, , Solution:, , Question 10:
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Solution:, , Question 11:, , Solution:
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Question 12:, , Solution:, , Question 13:, , Solution:
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Question 14:, , Solution:, , Question 15:, , Solution:, Put,, , Question 16:, , Solution:, Put,
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Question 17:, , Solution:, Put,, , Question 18:, , Solution:, Put,, , Question 19:, , Solution:, , Dividing Nr and Dr by, , , we get
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Let, , Question 20:, , Solution:, Put,, , Question 21:, , Solution:, Put,
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Question 22:, , Solution:, Put,, , Question 23:, , Solution:, Put,
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Question 24:, , Solution:, , Question 25:, , Solution:
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Question 26:, , Solution:, Let, , Question 27:, , Solution:, Put,, So,, , Question 28:, , Solution:, Put,
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Question 29:, , Solution:, Let, , Question 30:, , Solution:, Put,
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Question 31:, , Solution:, Put,, , Question 32:, , Solution:, Let I
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Question 33:, , Solution:, Put, I, , Question 34:, , Solution:, , Question 35:
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Solution:, Put,, , Question 36:, , Solution:, , Question 37:, , Solution:, Put,, , Let
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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,
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Thus, the correct option is B.
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EXERCISE 7.3, Find the integrals of the functions in Exercises 1 to 22:, Question 1:, , Solution:, , Question 2:, Solution:, Using,, , Question 3:
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Solution:, Using,, , Question 4:, , Solution:, Put, I
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Question 5:, , Solution:, Let I, , Let, , Question 6:, Solution:, Using,
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Question 7:, Solution:, Using,, , Question 8:, , Solution:
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Question 9:, , Solution:, , Question 10:
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Solution:, , Question 11:
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Solution:, , Question 12:, , Solution:
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Question 13:, , Solution:, , Question 14:, , Solution:, , Let
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Question 15:, , Solution:
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Question 16:, , Solution:, , Consider, Let, , From equation (1), we get, , Question 17:, , Solution:
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Question 18:, , Solution:, , Question 19:, , Solution:, , Let
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Question 20:, , Solution:, , Let, , Question 21:, Solution:, Let, Then,
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Let, , We know that,, , Substituting in equation (1), we get
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Question 22:, , Solution:, , Choose the correct answer in Exercises 23 and 24., Question 23:, is equal to, , Solution:, , Thus, the correct option is A.
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Question 24:, equals, , Solution:, , Put,, , Thus, the correct answer is B.
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EXERCISE 7.4, Integrate the functions in Exercises 1 to 23, Question 1:, , Solution:, Put,, , Question 2:, , Solution:, Put,, , Question 3:, , Solution:, Put,
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Question 4:, , Solution:, Put,, , Question 5:, , Solution:, Let
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Question 6:, , Solution:, Put,, , Question 7:, , Solution:, , For, , let
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From (1), we get, , Question 8:, , Solution:, Put,, , Question 9:, , Solution:, Put,
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Question 10:, , Solution:, , Let, , Question 11:
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Solution:, , Question 12:, , Solution:, , can be written as, Thus,
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Question 13:, , Solution:, , can be written as, , Thus,, , Let, , Question 14:, , Solution:
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Thus,, , Question 15:, , Solution:, Thus,
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Let, , Question 16:, , Solution:, , Equating the coefficients of x and constant term on both sides, we get
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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:
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Solution:, Let, Equating the coefficients of x and constant term on both sides, we get, , Let, , can be written as, Thus,
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Substituting equations (2) and (3) in equation (1), we get, , Question 19:, , Solution:
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Put,, Equating the coefficients of x and constant term, we get, , Let, , Then,, , and
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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
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Then,, , Using equations (2) and (3) in (1), we get, , Question 21:, , Solution:
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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
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Let, , Then,, Put,, , Substituting (2) and (3) in (1), we get, , Question 23:
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Solution:, Let, Equating the coefficients of x and constant term, we get, , Let, , Then,, , Put,, , Using equations (2) and (3) in (1), we get
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Choose the correct answer in Exercises 24 and 25., Question 24:, equals, , Solution:, , Hence, the correct option is B., Question 25:, equals
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Solution:, , Hence, the correct option is B.
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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
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Question 3:, , Solution:, Let, Equating the coefficients of, , Solving these equations, we get, , Question 4:, , Solution:, Let, , and constant terms, we get
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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
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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
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Consider, , , let, , Question 8:, , Solution:, Let, Equating the coefficients of, , and constant term, we get, , On solving these equations, we get
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Question 9:, , Solution:, , Let, , Equating the coefficients of, , and constant term, we get, , On solving these equations, we get, , Question 10:, , Solution:, , Let
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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
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Question 12:, , Solution:, On dividing, , by, , , we get, , Let, Equating the coefficients of, On solving, we get, , Question 13:, , and constant term, we get
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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
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Question 15:, , Solution:, , Let, , Equating the coefficients of, , On solving, we get, , and constant term, we get
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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, , ]
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Question 17:, [Hint: Put, , ], , Solution:, Put,, , Let, Equating the coefficients of t and constant, we get, On solving, we get
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Question 18:, , Solution:, , Let, , Equating the coefficients of, , On solving these equations, we get, , Question 19:, , and constant term, we get
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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
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Equating the coefficients of t and constant, we get, , Question 21:, , [Hint: Put, , ], , Solution:, Put, , Equating the coefficients of t and constant, we get
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Question 22:, equals, , Solution:, Let, Equating the coefficients of x and constant, we get, , Thus, the correct option is B.
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Question 23:, , equals, , Solution:, Let, Equating the coefficients of, , Thus, the correct option is A., , and constant terms, we get
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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
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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
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Question 6:, , Solution:, Let, Taking, , and, , and integrating by parts, we get, , Question 7:, , Solution:, Let, Taking, , and, , and integrating by parts, we get
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Question 8:, , Solution:, Let, Taking, , and, , and integrating by parts, we get
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Question 9:, , Solution:, Let, Taking, , and, , and integrating by parts, we get
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Question 10:, , Solution:, Let, , Taking, , and, , and integrating by parts, we get
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Question 11:, , Solution:, Let, , Taking, , Question 12:, , Solution:, Let, , and, , and integrating by parts, we get
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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
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Question 15:, , Solution:, Let, Let, , Where,, , Taking, , and, , Taking, , and, , Using equations, , and integrating by parts, we get, , and, , and integrating by parts,, , in, , ,
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Question 16:, , Solution:, Let, Let, , Question 17:, , Solution:
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Question 18:, , Solution:, , Let, It is known that,, From equation, , Question 19:, , Solution:, Let, Here,, It is known that,, , , we get
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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
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Question 22:, , Solution:, Let
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Using integration by parts, we get, , Question 23:, equals, A., , B., C., D., , Solution:, Let, Also, let, , Thus, the correct option is A.
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Question 24:, equals, A., B., C., D., , Solution:, Consider,, Let, , It is known that,, , Thus, the correct option is B.
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EXERCISE 7.7, Integrate the functions in Exercises 1 to 9., Question 1:, , Solution:, Let, , Since,, , Question 2:, , Solution:, , Since,
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Question 3:, , Solution:, , Question 4:, , Solution:, Consider,
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Question 5:, , Solution:, , Question 6:, , Solution:, , Question 7:, , Solution:, Put,
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Question 8:, , Solution:, , Question 9:, , Solution:
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Question 10:, , is equal to, , Solution:, , Thus, the correct option is A., Question 11:, , is equal to, , Solution:, , Thus, the correct option is D.
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EXERCISE 7.8, , Evaluate the following definite integrals as limit of sums., Question 1:, , Solution:, Since,, Here,, , where
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Question 2:, , Solution:, Let, , Since,, , , where, , Here,, , Question 3:, , Solution:, Since,, , where, Here,
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Question 4:, , Solution:, Let, , Let, , Since,, For,, , where, , , where
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For
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From equations (2) and (3), we get, , Question 5:, , Solution:, Let, , Since,, , Here,, , , where
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Question 6:, , Solution:, Since,, , , where, , Here,
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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
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Using second fundamental theorem of calculus, we get, , Question 4:, , Solution:, , Using second fundamental theorem of calculus, we get
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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
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Question 8:, , Solution:, Let, Using second fundamental theorem of calculus, we get, , Question 9:, , Solution:, Let, , Using second fundamental theorem of calculus, we get
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Question 10:, , Solution:, Let, Using second fundamental theorem of calculus, we get, , Question 11:, , Solution:, , Using second fundamental theorem of calculus, we get
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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
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Using second fundamental theorem of calculus, we get, , Question 15:, , Solution:, Let, Put,, As, , and as, , Using second fundamental theorem of calculus, we get
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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
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Question 17:, , Solution:, Let, , Using second fundamental theorem of calculus, we get, , Question 18:, , Solution:, Let, , Using second fundamental theorem of calculus, we get
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Question 19:, , Solution:, Let, , Using second fundamental theorem of calculus, we get, , Question 20:, , Solution:, Let
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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.
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Question 22:, , Solution:, , Put, , Using second fundamental theorem of calculus, we get, , Thus, the correct option is C.
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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
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Question 3:, , Solution:, Consider,, Let, , When, , Taking, , and, , and integrating by parts, we get
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Question 4:, , Solution:, Put,
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Question 5:, , Solution:, , Put,, , When, , Question 6:, Solution:, , Let
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Question 7:, , Solution:, , Put,, , When
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Question 8:, , Solution:, , Put,, When, , Let, , Then,, , Question 9:, The value of the integral, A. 6, B. 0, C. 3, D. 4, , is
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Solution:, Consider,, Let, and when, , When, , Put, , and when, , When, , Thus, the correct option is A., Question 10:, If, , , then, , is
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Solution:, Using integration by parts, we get, , Thus, the correct option is B.
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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,
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Adding (1) and (2), we get, , Question 3:, , Solution:, , Let, , Adding (1) and (2), we get
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Question 4:, , Solution:, Consider,, , Adding (1) and (2), we get, , Question 5:, , Solution:, Let, , As,, , on, , and, , on
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Question 6:, , Solution:, Consider,, , As, , Question 7:, , Solution:, , Consider,
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Question 8:, , Solution:, , Let
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Question 9:, , Solution:, , Consider,, , Question 10:, , Solution:, Consider,, , Since,, Adding (1) and (2), we get
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Question 11:, , Solution:, Let, , , therefore, , As, , If, , is an even function, then, , is an even function.
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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.
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Question 14:, , Solution:, Let, We know that,, , , if, , if, , Question 15:, , Solution:, Consider,, , Adding (1) and (2), we get
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Question 16:, , Solution:, Consider,, , Adding (1) and (2), we get, , Adding (4) and (5), we get
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Question 17:, , Solution:, Let, , We know that,, , Adding (1) and (2), we get, , …(1)
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Question 18:, , Solution:, Since,, , Question 19:, Show, , that, , Solution:, Let, , Adding (1) and (2), we get, , if, , f, , and, , g, , are, , defined, , as
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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
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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
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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
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Question 2:, , Solution:, , Question 3:, , Solution:, , Let
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Question 4:, , Solution:, , Multiplying and dividing by, , Let, , , we get
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Question 5:, , Solution:, , Let
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Question 6:, , Solution:, Consider,, , Equating the coefficients of, , and constant term, we get, , On solving these equations, we get, , From equation (1), we get
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Question 7:, , Solution:, , Put,
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Question 8:, , Solution:, , Question 9:, , Solution:, , Put,
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Question 10:, , Solution:, , Question 11:, , Solution:, , Multiplying and dividing by, , , we get
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Question 12:, , Solution:, , Put,
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Question 13:, , Solution:, , Put, , Question 14:, , Solution:, , Equating the coefficients of, , On solving these equations, we get, , and constant term, we get
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From equation (1), we get, , Question 15:, , Solution:, Let, , Question 16:, , Solution:, , Let
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Question 17:, , Solution:, , Put,, , Question 18:, , Solution:, , Put,
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Question 19:, , Solution:, Let, , As we know that,, , Let, Also, let
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From equation (1), we get, , Question 20:, , Solution:, , Put,
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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
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Question 23:, , Solution:, , Let
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Question 24:, , Solution:, , Let, , Using integration by parts, we get
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Question 25:, , Solution:, , Let, , Question 26:, , Solution:, Let
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Put,, When, , Question 27:, , Solution:, Consider,, , and when
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Consider,, Put,, When, , and when, , Therefore, from (1), we get
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Question 28:, , Solution:, Consider,, , Let, , When, , and when, , As, , , therefore,, , We know that if, , is an even function, then, , is an even function
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Question 29:, , Solution:, Consider,, , Question 30:, , Solution:, Consider,, Put,, , When, , and when
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Question 31:, , Solution:, Consider,, Put,, When, , Consider, , and when
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Question 32:, , Solution:, Let, , Adding (1) and (2), we get
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Question 33:, , Solution:, Consider,, , Where,
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From equations (1), (2), (3) and (4), we get, , Question 34:, , Solution:, Consider,, , Let,
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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.
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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
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Question 38:, , Solution:, Consider,, , Hence proved., Question 39:, , Solution:, Let, Using integration by parts, we get, , Put,, When, , Hence proved., , and when
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Question 40:, , as a limit of a sum., , Evaluate, , Solution:, Let, We know that,, , Where,, , and, , Here,, , Question 41:, , is equal to
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Solution:, , Consider,, , Put,, , Thus, the correct option is A., Question 42:, , is, , Equals to, Solution:, , Consider,, , Let
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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
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B. 0, C. -1, D., , Solution:, Consider,, , Adding (1) and (2), we get, , Thus, the correct option is B.