Exercises in Mathematical Analysis I

Transcription

1 Università di Tor Vergata Dipartimento di Ingegneria Civile ed Ingegneria Informatica Eercises in Mathematical Analysis I Alberto Berretti, Fabio Ciolli

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3 Fundamentals Polynomial inequalities Solve the following inequalities for R: Es ( )( 4) > 0 <, > 4 Es ( )( 3)( + ) < 0 < <, > 3 Rational inequalities Solve the following inequalities for R: Es 3 Es 4 Es 5 Es < < 0, 3 < < 5, > 8 < < < 0 < 3, 3 < <, > ( a)( b) a 0, a > b > 0 < a, b < a, > a 3 Irrational inequalities Solve the following inequalities for R: Es 3 > Es 8 8 < 9 + 4, Es 9 Es 0 Es Es Es 3 < + < > no solution

4 A Berretti, F Ciolli Eercises in Mathematical Analysis I Es 4 Es 5 Es 6 Es Es 8 Es 9 4 < < > < 9, > < < 4 + < 5 + R < < < 0 < 4 4 Absolute value inequalities Solve the following inequalities for R: Es 0 { } { 4} Es < 3 R Es 3 R Es { } { 3 + } { } 33 3 { Es 4 + < 3 < < 5 } Es 5 3 { < + 3} { > + } 5 Eponential and logarithmic inequalities Solve the following inequalities for R: Es < 8 Es 3 5 ( ) 4 5 ( ) + > 0 Es 8 log 3 ( + 03) > Es 9 log 5 ( + ) < 0 Es 30 log 0 ( + 4) > log 0 (3 + 0) < log 3 log 08 < log 3 log 5, > < < 5, + 5 R < < < <, > 3 4

5 A Berretti, F Ciolli Eercises in Mathematical Analysis I Es < 0 0 < < 6 Es > < < Es 33 log 0 (3 + 4) log 0 < 0 < < 6 Trigonometric inequalities Solve the following inequalities for R: π Es 34 sin cos > kπ < < π + kπ, π + kπ < < 5 3 π + kπ, k Z Es 35 cos + 3 sin Es 36 3 tan 4 3 tan + 3 > 0 π 6 + kπ 5 6 π + kπ, k Z π + kπ < < π 6 + kπ, π 3 + kπ < < π + kπ, k Z Es 3 log a ( sin ) < 0, a > 6 π + kπ < < 6 π + kπ, 5 6 π + kπ < < 6 π + kπ, k Z Es 38 3 cos + sin 3 > 0 not possible ( Es 39 4 cos + π ) 3 cos π + kπ π 6 + kπ, k Z cos Es 40 π sin 6 + kπ 5 6 π + kπ, 6 π + kπ 6 π + kπ, k Z Es 4 tan cot < kπ < < π 6 + kπ, 5 6 π + kπ < < π + kπ, k Z Boundedness of numerical sets Study the boundedness of the following numerical sets, epressing for any of them sup, inf, ma and min by verifying the definition { } Es 4 A = n +, n N inf A = 0, ma A = { } ( ) n Es 43 A = n +, n N min A = 3, ma A = { } + Es 44 A = 3, R, > 3 inf A =, sup A = + { } + Es 45 A =, R, < inf A =, sup A = 5

6 A Berretti, F Ciolli Eercises in Mathematical Analysis I { } nm Es 46 A = n + m, (n, m) N N \ {(0, 0)} { } nm Es 4 A = n + m, (n, m) N \ {0} { } n + m Es 48 A = n m, n, m N, n m { n Es 49 A = m + m } n, n, m N \ {0} min A = 0, ma A = inf A = 0, ma A = inf A =, sup A = + inf A =, sup A = + Study the boundedness of the following numerical sets, epressing for any of them sup, inf, ma and min { } 3n + Es 50 A = n +, n N \ {0} min A = 4 3, sup A = 3 { } Es 5 A = + n, n N \ {0} min A = 3, sup A = { } n Es 5 A = n! +, n N \ {0} inf A = 0, ma A = 4 3 { } log n! Es 53 A =, n N min A = 0, ma A = log n! { } n Es 54 A = sin( + nπ/), n N inf A =, sup A = + n n + Es 55 A = n, n N \ {0} min A = + 3, sup A = 0 { Es 56 A = ( ) n n n + 3 } 5, n N min A = 5, sup A = 6 5 { Es 5 A = n + sin( nπ ), n N} min A = 0, sup A = + { ( ) } (n + )π Es 58 A = sin /(n+), n N min A =, ma A = Establish if the following numerical sets are bounded; find sup, inf, ma and min, if they eist { } Es 59 A = + n, n N, n { Es 60 A = R : + > } Es 6 A = { R : < } Es 6 A = { R : log(sin ) R} inf A = 0, ma A = 3 A = (, ) (, + ); inf A =, sup A = min A =, sup A = 8 3 A = { π + kπ, k Z}; inf A =, sup A = + 6

7 A Berretti, F Ciolli Eercises in Mathematical Analysis I Es 63 A = { R : 3 + < 9} Es 64 A = { R : 5 < } 5 min A =, sup A = min A = 3, sup A = 3 { } Es 65 A = ( )n n, n N \ {0} Es 66 A = 4 n +, n +, n N, n even n N, n odd min A =, ma A = inf A = 0, ma A = 4 Es 6 Define an infinite set using a non-monotone sequence such that 0 and will be the inf and sup of the set respectively Es 68 Find inf e sup of the areas of the surfaces of the rectangles with perimeter equal to 4a, for a a positive real number, different from zero 8 Domain of functions Determine the domain of the following functions and study the boundedness of such sets Then trace a qualitative graph of the functions themselves Es 69 f () = Es 0 f () = + Es f () = 4 + Es f () = log / ( ) Es 3 f () = 6 log /3 ( ) Es 4 f () = log ( 5) Es 5 f () = log 3 ( + ) log 3 Es 6 f () = log 3 ( + ) Es f () = log 3 ( + ) log 9 ( + ) + Es 8 f () = (+)/( 3 4)

8 A Berretti, F Ciolli Eercises in Mathematical Analysis I Es 9 f () = log 5 (6 4 6 ) ( ) Es 80 f () = cos + ( ) Es 8 f () = cos + ( ( ) Es 8 f () = cos /4 + ) Es 83 f () = sin + cos Es 84 f () = log 3 (sin + cos ) Es 85 f () = log 3 (sin + cos ) Es 86 f () = log 3 (sin + cos ) ( ) + Es 8 f () = arccos ( ) + Es 88 f () = arcsin Es 89 f () = ( log 4 (sin ) ) / Es 90 f () = 4 log ( +) ( + ) / Es 9 Indicated by D the domain of any function of the eercises in the paragraph 8, determine the set of the interior points D of D and the set of its boundary points D Moreover, say if such sets are oper or closed and study their boundedness Es 9 Determine the set of the images (range) for any function of the eercises in the paragraph 8, and the set of the accumulation points of such sets Es 93 Given two functions f, g : A R R, show the following implications: f, g increasing = f + g increasing; f, g decreasing = f + g decreasing; 3 f increasing and g strictly increasing = f + g strictly increasing; 4 f decreasing and g strictly decreasing = f + g strictly decreasing Es 94 Establish under which conditions the following implication is true: f, g increasing (or decreasing) = f g increasing (or decreasing) 8

9 A Berretti, F Ciolli Eercises in Mathematical Analysis I Es 95 Furnish an eample such that the result of the eercise 94 is, in general ie without further hypothesis, false Es 96 Show that if f : A R R is invertible, then f increasing (decreasing) = f increasing (decreasing) Es 9 Let f : A R R be such that 0 f (A) and increasing Determine if f or decreasing is increasing Es 98 Let f, g : A R R two injective functions Is the function f + g invertible? Es 99 Let f : X Y and g : V W and let moreover f (X) V If f and g are invertible functions, is the composition f g an invertible function? Es 00 Furnish three different eamples of functions f : X X such that f f 9 Invertibilità di funzioni 0 Invertibility of functions Study the invertibility of the following functions in their natural definition set Es 0 f () = + Es 0 f () = + log / Es 03 f () = + log 3 ( + ) Es 04 f () = Es 05 f () = + if > Es 06 f () = al variare di a R + a if + a if 0 Es 0 f () = for any a R if > 0 3 if Es 08 f () = for any a R a if < Es 09 Let f : X Y and g : V W be two invertible functions such that it is well defined the composed function g f Call f and g their inverses respectively, show that (g f ) = f g 9

10 A Berretti, F Ciolli Eercises in Mathematical Analysis I Verify that the following functions are invertible; then determine the inverse of any of them, specifying its domain Es 0 f () = + Es f () = ( ), 0 Es f () = log / ( 3 ) Es 3 f () = Es 4 f () = e + e + ( Es 5 f () = sin 3 ), 0 + Es 6 f () = arccos(log ) Es f () = tan( 3 + ), Es 8 f () = arctan( 3 + ) π < 3 + < 3 π Es 9 f () = arcsin( + ), < 0 0

11 Limits of one real variable funtions Check, using the definition, the following its Verify the definition of it in the following cases: Es 0 = Es + = 0 Es 3 ( + ) = Es 3 = 4 Es 4 0 = + Es Es 6 3 = 3 Es Es 8 sin = π/ tan = + (π/) Es 9 = 0 + Es 30 = Es log / = + Es 3 Es = sin + = 0 Calculation of its Calculate, if they eist real or infinite,the following its: Es 34 ( + ) 9

12 A Berretti, F Ciolli Eercises in Mathematical Analysis I Es 35 Es sin cos 0 Es 3 Es 38 Es ( + 4 ) 0 ( + ) log ( + ) + log 3 log 3 Es 40 0 sin 9/0 Es Es / Es 43 4/ 0 sin Es 44 0 cos 4 sin Es 45 0 cos Es 46 π/ (sin )/ Es 4 0 Es 48 sin + log 4 + cos log Es 49 3 ( + ) 0 Es 50 0 log / cos Es 5 + (sin )/ log Es 5 Es log 3 0 ( ) + 0 log3 e log4 e 0 0 0

13 A Berretti, F Ciolli Eercises in Mathematical Analysis I Es log 3 (log ) + Determine domain and image of the following functions, indicating if they are periodic and even or odd Es 55 f () = sin + cos Es 56 f () = log 3 (sin 3 cos 3 ) Es 5 f () = log / ( sin + cos ) Es 58 f () = 4 (sin +cos )/(sin cos ) Es 59 f () = sin 3 cos Es 60 f () = α sin, al variare di α R 3 ( + e ) Es 6 f () = arcsin e 3 Es 6 f () = 4 tan ( + ) tan( + ) 6 Es 63 f () = Es 64 f () = arctan 5 5 Draw a qualitative graph of the functions studied in the eercises 55, 58, 60, 63 and 64 above Calculate, if they eist real or infinite,the following its: + Es 65 ( + 5) 6 + Es 66 Es 6 (log( + ) log ) + ( 3 + e Es 68 0 log cos sin ) ( +)/( 3) 4 Es (sin ) +3 log Es 0 0 log( + sin ) sin + log e 3 e 3 Es 0 sin e 3 3

14 A Berretti, F Ciolli Eercises in Mathematical Analysis I ( sin + ) Es 0 0 ( ) log( + ) + sin + Es Es 4 e / 0 Es 5 sin( 5 /3 ) log ( + /5 sin 3 ) 3 5 Calculate, if they eist real or infinite,the following its: cos( ) Es 6 log ( Es sin π ) / log(3 ) 4 Es 8 0 Es / log e Es 80 + ( ) ( cos(/) +)/ cos(/) ( ) / Es log Es 8 Es 83 + ((e / + ) / cos(/)) + ((e / ) / cos(/)) e /(3 ) + e(4 3 cos( 3)) /5 e 4 Es 84 3 cos( 3) Es 85 Es 86 sin(/) + log( + e / + /(+) ) + Es e log log 0 ( + + ) sin ( 3 + log0 ( ) )0 arcsin cos 4 4 Es 88 0 ( + sin ) / arctan e 4

15 A Berretti, F Ciolli Eercises in Mathematical Analysis I Calculate, if they eist real or infinite,the following its: Es sin + log ( + e ) Es 90 Es 9 Es 9 Es 93 ( 4 e + sin(/ ) + ) ( + 3 ) ( ) 4 + log Es arctan e 3 π arctan π/ ( + ) π/+/ 0 log arctan π/ ( + ) π/ e / + + log + log ( e / + e /) + sin cos + e / Es cos arcsin e + ( + ) tan( ) sin 3 ( ) Es 96 0 cos( ) ( cos 3 ) + sin 3/4 Es e / + ( e ) + ( cos 3 ) + sin 3/4 Es e / + ( e ) / Es e 3 Es 00 log log + log log ( 0 + log ) e cos( ) + log (sin π/) Calculate, if they eist real or infinite,the following its: 4 ( e π ) Es ; 0 + ; 0 + ; 0; n times {}}{ Es if n is even, 0 if n is odd 5

16 A Berretti, F Ciolli Eercises in Mathematical Analysis I Es 03 Es 04 ( + ( ) sin(π)(e e ) )log cos(/) e / ( 4 ) log + + sin(e cos + sin + sin ) Es sin 4 e cos + sin 3 + sin 3 Es sin sin cos Es tan 6 (/) log( cos )( cos ) / sin e Es 08 0 sin log(cos )( ) Es e / cos ( Es 0 (sin + ) log(sin + ) ) ( +3 )/ 0 0 ( ) e / log sin 3 (/) Es ( ) log 3 Arrange in growing order of infinity (infinitesimal) the following functions and sequences, after having determined the order of infinite (infinitesimal), if it eits as a real number Es For + : a) e, b) log, c) log, d) sin(/) d, b, c, a ord d= Es 3 For n + : a) n, b) n!, c) n n, d) ( 3 n a, d, b, c ) Es 4 For + : a), b) log, c) log, d) Es 5 For 0 + : a) log, b), c) Es 6 For 0 + : a) log, b) log log, c) log + b, d, c, a ord d= 3 cos, d) log arcsin arcsin a, c, d, b ord b=, c= 6 log, d) log( + ) b, a, c, d ord d= 6

17 A Berretti, F Ciolli Eercises in Mathematical Analysis I Es For + : a) e /( ), b) 0 cos( ), c) sin 3 3, d) log 0 ( ) c, d, b, a ord b=, c= 3 Es 8 For + : a) d) ( ) /( ) Es 9 For 0 + : a) arctan, ( ) 3/, b) ( ) 3/4 log( ), c) e / sin( ), a, b, c, d ord a= 3 b) cos log, c), d) sin 3 4 d, c, b, a ord a=, d= 3 4 Arrange in growing order of infinity (infinitesimal) the following functions and sequences, after having determined the order of infinite (infinitesimal), if it eits as a real number Es 0 For + : a), b) log( + 3 ( ) +/ log + e 3 ), c) +, d) + c, d, a, b ord a=, b=3, c= Es For n + : a) n n +, b) n log n, c) log n n, d) n! (n + )! (n )! c, d, b, a ord a= 3, d= Es For n + : a) ( n n ), b) n( 3 + n n), c) (cos(/n) ) n3 /(n+), d) n n a, b, c, d ord b= Es 3 For 0 + : a) ( cos ) log( + sin 4, ) b) log(+), c) log, d) sin( log(+)) log c, b, d, a ord a=, b= Es 4 For + : a) log( cos(/)) sin, b) ( ) (/) 00, c) + log, d) log 00 ( + ) d, a, c, b ord a= ( 3) Es 5 For 3 + : a) (e (3 )(+) 3 ) sin( 3) 9/4, b) sin 3 ( 3), c) ( 3) 3 log( ), d) ( 3) 3 log 0 ( 3) d, b, a, c ord a= 3 4, b=3, c=4 3 Es 6 For 0 + : a) log( + ), b) /( +), c) 5 + 4, d) 3 log 0 + b, c, d, a ord a=3, b=, c=4, c= 5 Es For 0 + : a) arctan, b) ( cos ( ) + ) 4 + log( + ), c) + log e, d) sin( 3 log ) a, c, b, d ord a= 3, b= Calculate the it of the following sequences:

18 A Berretti, F Ciolli Eercises in Mathematical Analysis I Es 8 Es 9 Es 30 Es 3 Es 3 Es 33 e n n + n n n + n + e n3/ n n + e n n + n 3 n + sin n 4 + n 5 + n 6 (+log/ n) n + n / n + (log(n + ) log n log(n + )) + n n 3 n n + 4 n Calculate the it of the following sequences: Es 34 Es 35 Es 36 Es 3 Es 38 Es 39 Es 40 Es 4 Es 4 Es 43 n + n + n + n + n (n + ) sin(/n) n n (n!)! n n! log n n + n+ n+ n ( + n! ) (n )nn (n+)! n + n n n + n + arcsin ( e n e e ) n 0 + n ( + cos(/n) n + cos(/n)) (arcsin(/n))n ( n + n n + n + n + n 3 e /n ) n n 6 + e n log n + n4 arcsin(/n) n + n n n! + e n3 n + n n e n e Es 44 Es 45 n + n + n e n + sin(πn/) n + sin n e 8

19 A Berretti, F Ciolli Eercises in Mathematical Analysis I *Es 46 Let {a n } be a positive terms sequence such that log a n 0 n + a n+ Give at least two countereamples showing that from this relation is not possible deduce that a n = + n + Moreover, say under which further hypothesis the result would be true Es 4 Using the comparison theorem, show that n + n + + n n + n = 0 *Es 48 Let {a n } be a positive terms sequence Show that a n+ = r 0 n + a n n + n an = r Use the sequence a n = e /n + sin(πn/) + to show that in general the converse is not true *Es 49 Show, ehibiting a countereample, that if {a n } is a non-negative terms sequence, then a n n + a/n n = l n + l n = Moreover, show that if a /n n = l > then a n + for n + n 9

20 3 Study of functions of one real variable 3 Asymptotes Determine the possible asymptotes (vertical, horizontal, oblique) for the following functions, after having indicated their domain Moreover, calculate the it of the functions to the boundary points of their domain Es 50 f () = + 3 Es 5 f () = ( ) Es 5 f () = 4 + Es 53 f () = log( + ) Es 54 f () = Es 55 f () = + Es 56 f () = arcsin + Es 5 f () = e log (/( ))+log(3 3)+ Es 58 f () = log( 3e + e ) Es 59 f () = e /( ) Es 60 f () = cos + Es 6 f () = log 3 + log + + Es 6 f () = arctan (Use the formula arctan + arctan = π, > 0) Es 63 f () = +/ log Es 64 f () = +log / +log Es 65 f () = 4 e / 0

21 A Berretti, F Ciolli Eercises in Mathematical Analysis I 3 Continuity and derivability Determine the domain and the set of continuity of the following functions Es 66 f () =,, > Es 6 f () = + Es 68 f () = 4 / sin sin(log ) Es 69 f () = log Es 0 f () = sin(cot ), kπ, k Z 0, = kπ, k Z Es Determine a R such that the following function result to be continuous f () = +, a, = Es Say if it is possible to apply the Weierstass theorem about the eistence of the etremes to the following function f () =, 0 <, 3 Determine the set of continuity and the set of derivability of the following functions and calculate their derivative Es 3 f () = tan Es 4 f () = e e Es 5 f () = 3 Es 6 f () = + Es f () = Es 8 f () = 4 Es 9 f () = +

22 A Berretti, F Ciolli Eercises in Mathematical Analysis I Es 80 f () = (arcsin ) 3 Es 8 f () = e sin ( ) Es 8 f () = arctan Es 83 f () = log tan ( ) Es 84 f () = arcsin + ( ) Es 85 f () = arcsin Es 86 f () = e /( ) Es 8 f () = arccos 3 Es 88 f () = log Es 89 f () = log 3 log() Es 90 f () = + e Es 9 f () = + 4 arctan Es 9 f () = cos ( ) Es 93 f () = log The same work (determination of continuity, derivability and calculation of the derivative) is recommended also for the functions in the eercises in paragraphs 8 and 3 33 Invertibility and derivative of the inverse function Verify the invertibility of the following functions and determine the domain of derivability of the respective inverse functions Es 94 f () = + log Es 95 f () = + e Es 96 f () = + log( + ) Es 9 f () = + sin

23 A Berretti, F Ciolli Eercises in Mathematical Analysis I Es 98 f () = + arctan Es 99 f () = 5 cos Es 300 For any of the following function f (), determine: f (), f (), f ( + log ), f ( π + ), f ( + π 4 ), f (0) Moreover, write the equation of the tangent line passing for the point indicated Es 30 Use the mean value theorem to show that sin sin y y,, y R 34 Critical points Determine the possible critical points for the following functions Es 30 f () = Es 303 f () = + Es 304 f () = log Es 305 f () = e / Es 306 f () = + log Es 30 f () = log Es 308 f () = 3 + Es 309 f () = ( + ) Es 30 f () = e ( 3 + (3 8) ) Es 3 f () = (( ) 6 ) log 35 Derivability and Monotony Determine the intervals of monotony for the functions in the paragraph 34 3

24 A Berretti, F Ciolli Eercises in Mathematical Analysis I 36 Taylor and Mac Laurin Polynomials Determine the Mac Laurin polynomial of the following functions to the indicated order Es 3 f () = sin( ), to the order 4 Es 33 f () = +, to the order 3 Es 34 f () = log( + 3 ), to the order 8 Es 35 f () = sin, to the order 4 Es 36 f () = e +, to the order 5 Determine the Taylor polynomial, centered in 0 and to the indicated order, for the following functions Es 3 f () = e, 0 =, to the order 3 Es 38 f () = cos, 0 = 3, to the order 4 Es 39 f () = log( + ), 0 =, to the order 3 Es 30 Determine the Mac Laurin polynomial of order 4, for the function f () = log( + sin ) Determine the Mac Laurin polynomial of order 5, for the following functions Es 3 f () = ( + )e Es 3 f () = sin + cos Es 33 f () = sin log( + ) 3 Using Taylor polynomials for the calculation of its Calculate the following its 3 ( Es 34 e /(+) ) + + sin + cos e4 Es log( + ) e /(+) + log Es 36 + ( ) log ( cos( )) + Es log ( arctan log π ) π 5 π 4 4

25 A Berretti, F Ciolli Eercises in Mathematical Analysis I 38 Uniform Continuity Es 38 Verify, using the definition, that f () = is not an uniform continuous function over X =, + ) Es 39 Establish if f () = arctan is a uniform continuous function over the following domains: D = (0, + ); D = (, + ); D 3 =, + ); D 4 = (, ) (, + ) Es 330 Verify if f () = log results to be a Lipschitz function over the domain D =, + ) Verify if the following functions result to be uniformly continuous over their domain: Es 33 Es 33 e /, if 0, f () = 0, if = 0 sin +, if < 0, f () = log(e( + )), if 0 Es 333 f () = sin(e sin ) For any of the following functions determine a R such that they result to be continuous Then check if, for such an a, the functions result to be also uniformly continuous throughout their domain of definition Es 334 Es 335 Es 336 a(e ), if <, f () = e, if log + e, if >, f () = a, if =, π + arctan, if < 4 +, if 0, f () = a log( + ), if > 0 5

26 A Berretti, F Ciolli Eercises in Mathematical Analysis I Es 33 sin, if > 0, log( + ) + f () = a, if = 0, (e + ), if < 0 6

27 4 Integrals of one-variable functions and numerical series 4 Immediate indefinite integrals (primitives) Calculate the following indefinite integrals (primitives) Es d 3 Es 339 Es 340 Es 34 Es 34 Es 343 3q d, q R + (a /3 /3 ) 3 d, a R P n () d, P n () = n a k k, a k R k=0 n α k e βk d, α k, β k R, β k 0 k=0 n α k sin β k d, α k, β k R, β k 0 k=0 3 + Es 344 d 3 + Es d 4 /4 + c 3 (3q)/ 3/ + c a 9 5 a4/3 5/3 + 9 a/3 / c n a k k + k+ + c k=0 k=0 n α k e βk + c β k k=0 n α k cos β k + c β k 3 + log + c c Es 346 a + d, a R a arcsin + + c Es 34 Es 348 Es 349 d arctan + c + tan d tan + c cot d cot + c Es ( + ) d + arctan + c

28 A Berretti, F Ciolli Eercises in Mathematical Analysis I Es 35 Es 35 Es 353 Es 354 Es 355 Es 356 Es 35 Es 358 sin d cos + c cos d c n a n n d, a R a n + a n n + a n n + + an + c d sin tan cot + c cos cos d cos + sin + c sin + cos sin d ( sin ) + c cos 3 d sin 3 + c sin cos d tan cot + c 4 Indefinite integrals by substitution Calculate the following indefinite integrals using, for instance, the method of substitution of variable sin Es 359 cos d 3 sin3/ + c Es 360 d log + c Es 36 a + d, a R+ a arctan a + c Es 36 a d, a R+ a log a + c a + a Es 363 d, a R + a arcsin a + a + a + c Es 364 Es 365 Es e + e d log + e + c a b d, a, b R+ b arcsin b a + c d + c 8

29 A Berretti, F Ciolli Eercises in Mathematical Analysis I Es 36 Es 368 Es 369 Es 30 Es 3 Es 3 Es 33 Es 34 Es 35 Es 36 Es 3 Es 38 Es 39 Es 380 Es 38 Es 38 Es 383 Es d 5 + arctan + c sin α cos d, α α + sinα+ + c e + e d arctan e + c cos(log ) d sin(log ) + c + d ( arctan ) + c ( ) 3 d log ( ) + c a 4 d, a 0 arcsin 4 a + c cot sin α d, α R+ α sin α + c d, a 0 a a arctan a + c a d, a R a a arccos a + c a + b c + d d, a, b, c, d R, c 0 c ( a(c + d) + (bc ad) log c + d ) + cons e e d (e 3 ) 3 + e + + c tan log(cos ) d log(cos log ) + c a (a + )(a d, a 0 ) / a a + + c sin cos d log tan + c sin d sin log + cos + c cos d log + sin cos + c a d, a R + arctan (a + ) a + c 9

30 A Berretti, F Ciolli Eercises in Mathematical Analysis I Es 385 Es 386 Es 38 Es 388 *Es 389 log a + d, a R a c (a + d, a 0 ) 3/ (a + d, a 0 ) 5/ (a + d, a 0 ) / a 6 ( a + 3 a 4 ( a (a + ) a a + + c 3 (a + ) 3 ) + c 5 (a + ) 5 ) + c (a + ) d, a 0, n N use the results of the previous eercises (n+)/ 43 Indefinite integrals by parts Calculate the following indefinite integrals using, for instance, the method of integration by parts Es 390 Es 39 Es 39 Es 393 log d 3 sin 3 d sin 4 d sin 5 d (log + ) + c ( sin cos + cos ) + c sin + 3 sin 4 + c cos + 3 cos3 5 cos5 + c Es 394 Es 395 Es 396 Es 39 Es 398 Es 399 sin e d ( sin e + cos e ) + c 3 4 arctan d 4 arctan + ( ) 3 3 arctan + c log α+ + c if α 0, ; log + c, if α = 0; log log + c if α = α + e d e e + c e d n e d, n N e ( + ) + c e ( n n n + n(n ) n + ( ) n n!) + c 30

31 A Berretti, F Ciolli Eercises in Mathematical Analysis I Es 400 sin d cos + sin + c Es 40 cos d sin + cos + c Es 40 sin d cos + sin + cos + c Es 403 cos d cos + sin + cos + c Es 404 I n = n sin d, n N, n > Let I = cos d, I n = n cos + ni n = = n cos + n( n cos + (n )( n cos + + cos d) ) Es 405 I n = n cos d, n N, n > Let I = sin d, I n = n sin ni n = = n sin n( n sin (n )( n sin sin d) ) Es 406 Es 40 Es 408 Es 409 Es 40 Es 4 Es 4 Es 43 Es 44 Es 45 sin d d arcsin d ( ) sin cos + + sin + c ( + arcsin ) + c arcsin + ( 4 arcsin ) + c cos d tan + log cos + c arcsin d e arcsin d sin p cos q d, p, q R, p q + a d, a R e sin d e cos d arcsin + arcsin + c earcsin ( + ) + c q p q sin p sin q + p q cos p cos q + c p ( + a + a log + a + + c e (sin cos ) + c e (sin + cos ) + c 3

32 A Berretti, F Ciolli Eercises in Mathematical Analysis I Es 46 Es 4 Es 48 Es 49 *Es 40 (Use the formula: e α sin d, α R e α cos d, α R e α sin β d, (α, β) R, (α, β) (0, 0) e α cos β d, (α, β) R, (α, β) (0, 0) e cos n d, n N cos n = n n n/ k=0 ( n n/ ) ( ) n cos(n k), k + n n/ k=0 α + eα (α sin cos ) + c α + eα (sin + α cos ) + c α + β eα (α sin β β cos β) + c α + β eα (β sin β + α cos β) + c ( ) n cos(n k), k n odd n even and the result of the eercise 49) *Es 4 e sin n d, n N (Use the formula: sin n = n n n/ ( ) n ( ) n/ k sin(n k), k k=0 ) + n/ ( ) n n ( ) n/ k sin(n k), k ( n n/ k=0 n odd n even and the result of the eercise 48) *Es 4 I m,n = sin m cos n d, m, n Z (One obtains the following equivalent reduction formulas: I m,n = sinm cos n+ n + *Es 43 = sinm+ cos n+ m + + m n + I m,n+ = sinm+ cos n + n m + m + I m+,n = e α sin m β d, α, β R, m Z (Use the results of the previous eercises) *Es 44 e α cos n β d, α, β R, n Z (Use the results of the previous eercises) + m + n + m + I m+,n = sinm+ cos n+ n + + m + n + I m,n+ ) n + 3

33 A Berretti, F Ciolli Eercises in Mathematical Analysis I 44 Determine the following indefinite integrals (primitives) + 9 Es 45 ( 3) ( + ) d 5 log 3 5( 3) + 6 log + + c Es 46 ( )( ) 3 d log + log + ( ) 5 ( ) + c Es 4 3 d log + 5 log 3 log + + c Es 48 ( ( + )( ) d log( + ) ) arctan + log + c + Es 49 + d log( + ) + arctan + c 3 6 Es d 5 ( ) + + log log + c Es d 4 log log log arctan c Es 43 ( )( + 5) d log arctan c / Es 433 d log arctan c Es 434 ( ) ( + 5) d log arctan c 8 Es 435 Es 436 Es 43 Es 438 Es ( + )( + + ) d 3 + ( + ) d ( 3 + ) d ( + ) d d log log log log log arctan + 3 arctan + 3 arctan + 3 arctan + 3 arctan c + c + c + c + c 33

34 A Berretti, F Ciolli Eercises in Mathematical Analysis I Es 440 Es 44 Es 44 Es 443 Es 444 Es 445 Es 446 Es 44 tan tan 3 + d sin cos + sin d sin m cos n d, m, n N cos m sin n d, m, n N 4 + d log log log log log arctan + 3 arctan + 3 arctan + 3 arctan + 3 arctan d log arctan c tan tan d cos d log log arctan + 3 arctan c + c + c + c + c + c + c 34