Outside the high school program / Integration 47 exercises (including 33 corrected)

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ChingQuizz : 2 exercises available for Quizz assessment : 1. Integral by parts E.3297 Let f be the function defined on 0 ; + ∞ by: f ( x ) = x · ln x + 1 1 We pose : I = 1 0 x 2 x +1 d x a Determine three real a , b and c such that, for any x =1 : x 2 x + 1 = ax + b + c x + 1 b Calculate I . 2 Determine by integration by parts the following integral: 1 0 x · ln( x +1) d x E.3959 Using integration by parts, calculate the following integrals : a 4 − 1 x · e x d x b 5 0 t · e 2 · t d t c 1 e ln t d t d 1 0 (2 · x + 1) · ln x + 1 d x e e 1 5 · ln x x 2 d x f 1 1 2 x 2 · ln x d x E.3961 Using integration by parts, calculate the following integrals : a 2 0 ln( x +1) ( x + 1) 2 d x b ln 3 1 e t · ( t − 1) d t c π 6 0 x · sin 3 x d x d e 1 x · (1 − ln x ) d x E.4016 The function f is defined on the interval 0 ; + ∞ by: f ( x ) = 20 · x + 10 · e − 1 2 · x Calculate the mean value of the function f on the interval 0 ; 3 . E.4294 Consider the function f defined on the interval 0 ; + ∞ by: f ( x ) = ln x − 1 2 · x + 1 1 Calculate 1 1 2 ln( x ) d x . Integration by parts may be used. 2 Calculate, in units of area, the measure of the area of the domain bounded by: the representative curve of the function f , the x-axis; the straight lines of equation x = 1 2 and x =1 . (it will be assumed that the function f is positive on 1 2 ; 1 ) E.4295 Using integration by parts, show that : e 1 ln x x 2 d x = 1 − 2 e 2. Integral by parts - doubles E.3960 Using double integration by parts, cal-culate the following integrals : a 3 − 2 x 2 · e x d x b 5 1 1 − t 2 e − t d t E.4298 The plane provided with an or-thonormal reference frame O ; −→ i ; −→ j . Consider the function f defined on R by: f ( x ) = x 2 · e − x Let f be the derivative function of f . For any real number a , consider the integral: I ( a ) = a 0 f ( x ) d x 1 Give according to the values of a the sign of I ( a ) . 2 Using double integration by parts show that for any real number a : I ( a ) = 2 − 2 · e − a · 1 + a + a 2 2 3. Integral by parts - study https://chingmath.fr sacados/3297 chapExoCorrec/3959 sacados/3959 chapExoCorrec/3961 sacados/3961 chapExoCorrec/4016 sacados/4016 Extrait de France Septembre 2005 sacados/4294 sacados/4295 chapExoCorrec/3960 sacados/3960 sacados/4298
E.3983 Consider the function f defined on 1 ; + ∞ by: f ( x ) = x − 1 · e 1 − x We denote by ( C ) the representative curve of the function f in an orthonormal frame O ; −→ i ; −→ j . This curve is given in the appendix. For any real number x greater than or equal to 1 , we posit : F ( x ) = x 1 f ( x ) d t = x 1 ( t − 1) · e 1 − t d t 1 Demonstrate that the function F is increasing on 1 ; + ∞ . 2 Show, using integration by parts, that for any real x be-longing to 1 ; + ∞ : F ( x ) = − x · e 1 − x + 1 3 Show that on 1 ; + ∞ , the equation F ( x )= 1 2 is equiva-lent to the equation : ln(2 x ) + 1 = x E.4004 The plane is referenced to an or-thonormal frame O ; −→ i ; −→ j ; the graphical unit is 4 cm . Let f be the function defined on R by: f ( x )= e 2 x − 1 e 2 x +1 Γ is its representative curve in the reference frame O ; −→ i ; −→ j . 1 Show that f ( x )= e − x − e − x e x +e − x ; deduce a primitive of f . 2 What is the area in cm 2 of the surface between Γ , the straight line with equation y = x and the straight lines with equations x =0 and x =1 ? Hatch this surface on the graphical representation. 3 Calculate: 1 0 f ( x ) 2 d x 4 Using integration by parts, show that : 1 0 x · 1 − f ( x ) d x = e 2 − 1 e 2 + 1 − ln e 2 + 1 2 · e . Deduce : 1 0 x · f ( x ) 2 d x E.4296 Let f be the function defined for any real number x in the interval 0 ; 1 by: f ( x ) = 1+ x · ln x Let ¸ be a real number such that 0 <¸< 1 . We pose : I ( ¸ ) = 1 α 1 − f ( x ) d x 1 Using integration by parts, show that : I ( ¸ ) = ¸ 2 2 · ln ¸ + 1 4 − ¸ 2 4 2 Determine lim α ↦→ 0 I ( ¸ ) 3 Admit that the function f is positive on 0 ; 1 . Interpret the previous result graphically. 4. Integral by parts - sequences E.3982 For any natural number n 2 , consider the integral I n defined by: I n = 2 1 1 x n · e 1 x d x 1 Calculate I 2 . 2 A recurrence relation: a Demonstrate, using integration by parts, that for any natural number n 2 : I n +1 = e − √ e 2 n − 1 + (1 − n ) · I n b Calculate I 3 . E.3986 Consider the numerical sequence J n defined, for any non-zero natural number n , by: J n = n 1 e − t · 1 + t d t 1 Show that the sequence J n is increasing. 2 In this question, the candidate is invited to write on his copy the steps of his approach even if it is not successful. We define the sequence I n , for any non-zero natural number n , by: I n = n 1 ( t + 1) · e − t d t a Justify that, for any t 1 , we have : t + 1 t + 1 b Deduce that : J n I n . c Calculate I n as a function of n . Deduce that the se-quence J n is increased by a real number (independent of n ) . d What can we conclude from this for the sequence J n ? https://chingmath.fr sacados/3983 sacados/4004 sacados/4296 Extrait Antilles-guyane Septembre 2009 chapExoCorrec/3982 sacados/3982 Extrait d'Asie Juin 2010 chapExoCorrec/3986 sacados/3986
E.4005 For any natural number n , we pose : I n = π 0 e x · cos( n · x ) d x 1 Show that, for any natural number n : cos n · x = ( − 1) n ; ; sin n · ı = 0 2 Using two integrations by parts, show that : I n = ( − 1) n · e π − 1 1 + n 2 E.4006 We pose : I 1 = e 3 2 1 e ln x d x and I 2 = e 3 2 1 e ln x 2 d x . 1 Calculate I 1 . 2 Using integration by parts, show that : I 2 = 5 4 · e 3 2 − 5 e E.4222 For any natural number n 2 , con- sider the integral I n defined by: I n = 2 1 1 x n · e 1 x d x 1 Calculate I 2 . 2 Demonstrate, using integration by parts, that any natu-ral number n 2 : I n +1 = e − e 2 n − 1 + (1 − n ) · I n E.4303 For any natural number n 2 , con-sider the integral I n defined by: I n = 2 1 1 x n · e 1 x d x 1 Calculate I 2 2 Demonstrate, using integration by parts, that for any natural number n 2 : I n +1 = e − e 2 n − 1 + (1 − n ) · I n 3 Calculate I 3 . 5. Integral by parts - probability E.4269 We model the waiting time between two customers at a counter as a random variable following an exponential distribution with parameter – . The probability of a customer waiting less than t minutes is defined by: P ( X t ) = t 0 – · e − λ · x d x The average waiting time is given by: lim t ↦→ + ∞ t 0 – · x · e − λ · x d x 1 Using integration by parts, calculate t 0 – · x · e − λ · x d x as a function of t . 2 Deduce that the average time is 1 – 3 Since the average waiting time is 5 min , what is the prob-ability of waiting more than 10 min ? more than 5 min ? 6. Integral by parts - annals E.3124 The objective is to study some properties of the function f defined on the interval − 1 ; + ∞ by: f ( x ) = 1 − x 2 · e − x Part A : Variations of f and graph of the curve ( F ) Let f be the function defined on the interval − 1 ; + ∞ by: f ( x ) = 1 − x 2 e − x In the plane ( P ) equipped with the orthonormal coordinate system O ; −→ i ; −→ j (graphical unit : 2 cm ) the graphical rep-resentation of the function f is denoted by ( F ) . 1 Determine the limit at + ∞ of f : interpret this result graphically. 2 a Determine, based on the values of x in the interval − 1 ; + ∞ , the sign of x 2 − 2 x − 1 and that of f ( x ) . b Determine the derivative function f of f . Deduce the direction of variation of f and then draw up its table of variations.; Specify the exact values of the minimum and maximum. 3 Determine an equation for the tangent line ( T ) to the curve ( F ) at point A of ( F ) whose x-coordinate is 0 . 4 a Determine the exact value, then the value rounded to 0.1 , of each of the slope coefficients of the tangents to the curve ( F ) at (1 ; 0) and C ( − 1 ; 0) . b Draw the three tangents to the curve ( F ) at A , B ( A ; 0) , and C ( − 1 ; 0) and the curve ( F ) . Part B: Integrals and areas The areas S and S 1 ( u ) of the plane ( P ) , whereù u is a given real number from the interval 1 ; + ∞ are defined by: S is the set of points M ( x ; y ) such that : 0 x 1 et 0 y f ( x ) . S 1 ( u ) is the set of points M ( x ; y ) such that : 1 x u et f ( x ) y 0 . The respective areas of these surfaces are denoted by A , A 1 ( u ) . Their exact values will be expressed in units of area. 1 Justify the existence of the integral x 1 f ( t ) d t où x is a positive real number. Using two successive integrations by parts, determine https://chingmath.fr sacados/4005 sacados/4006 Extrait Antille-Guyane Septembre 2001 sacados/4222 Extrait d'Asie Juin 2010 sacados/4303 Extrait d'Asie Juin 2010 sacados/4269 Extrait d'Antilles-guyane Septembre 2005 chapExoCorrec/3124 sacados/3124
0x0∞2;32;4-∞0∞xVariationdeg P0M0(Cg(CfH0IJOij this integral. 2 Deduce the exact value of 0 1 f ( t ) d t . Deduce the exact value of the area A 3 Determine, as a function of u où u 1 , the area A 1 ( u ) and then the limit, when u tends towards + ∞ , of A 1 ( u ) . Interpret this result graphically. 4 The objective is to determine the real ¸ greater than or equal to 1 for which : A 1 ( ¸ ) = A a Demonstrate that, on the interval 1 ; + ∞ , the equa-tion A 1 ( x )= A is equivalent to : x =2 · ln(1+ x ) b Study the direction of variation of the function h de-fined on the interval 1 ; + ∞ by: h ( x )= x − 2 · ln(1+ x ) . Demonstrate that, on the interval 1 ; + ∞ , the equa-tion x =2 · ln(1+ x ) admits exactly one solution and that this solution, denoted ¸ , verifies the condition : 2 <¸< 3 . c Determine, indicating the method used, a frame of am-plitude 10 − 3 of ¸ . Determine f ( ¸ ) in the form of a rational function of ¸ and then the framework of f ( ¸ ) , to be deduced from the previous one, of amplitude 2 × 10 − 4 E.3177 1 Consider the function g defined on 0 ; + ∞ by: g ( x ) = ln x − 2 x The table of variations of g is given below : Demonstrate all the properties of the g function grouped in this table. 2 Let f be the function defined on 0 ; + ∞ by: f ( x ) = 5 ln x x a Show that f ( x 0 )= 10 x 2 0 where x 0 is the real appearing in the table above. b Let a be a real. For a> 1 , express a 1 f ( t ) d t as a func-tion of a . 3 O ; −→ i ; −→ j below the representative curves of the func-tions f and g noted respectively ( C f ) and ( C g ) . We call I the point with coordinates (1 ; 0) , P 0 the point of intersection of ( C g ) and the x-axis, M 0 the point of ( C f ) having the same abscissa as P 0 and H 0 the orthog-onal project of M 0 on the y-axis. We name ( D 1 ) the domain of the plane bounded by the curve ( C f ) and the segments [ IP 0 ] and [ P 0 M 0 ] . We name ( D 2 ) the plane domain bounded by the rectan-gle constructed from [ OI ] and [ OH 0 ] . Demonstrate that the two domains ( D 1 ) and ( D 2 ) have the same area, then give an amplitude frame 0.2 of this area. https://chingmath.fr chapExoCorrec/3177 sacados/3177 0x0∞2;32;4-∞0∞xVariationdeg P0M0(Cg(CfH0IJOij
-12345I-0.6-0.5-0.4-0.3-0.2-0.100.10.20.30.40.50.60.70.8 E.3211 Let f be the function defined on 0 ; + ∞ by: f ( x ) = 2 ln x x 2 + x 1 Show that for all x> 1 : ln x x 2 f ( x ) ln x x 2 a Calculate I = 4 2 ln x x d x and J = 4 2 ln x x 2 d x (we can use integration by parts for the latter) b Deduce a frame for K = 4 2 f ( x ) d x . 3 the figure below represents the representative curve of f (graphic units: x-axis 1 cm for 1 unit, y-axis 4 cm for 1 unit) . Consider the set of points M ( x ; y ) such that : 2 x 4 0 y f ( x ) and note A its area. Using the frame found at 2 b , give a frame A in cm 2 . E.3220 1 Let g be the function defined on the interval 1 ; + ∞ by: g ( x ) = 1 x ( x 2 − 1) a Determine the real numbers a , b and c such that for all x> 1 : g ( x ) = a x + b x + 1 + c x − 1 b Find a primitive G of g on the interval 1 ; + ∞ . 2 Let f be the function defined on the interval 1 ; + ∞ by: f ( x ) = 2 x x 2 − 1 2 Find a primitive F of f on the interval 1 ; + ∞ . 3 Using the results obtained previously, calculate: I = 3 2 2 x x 2 − 1 2 ln x d x We’ll give the exact result as p · ln2+ q · ln3 , with p and q rational. E.3236 Let f be the function defined on the interval [0 ; + ∞ [ by: f ( x ) = x − 1 2 − e − x Its representative curve C is plotted in the orthonormal frame below (graphic unit 2 cm ) . 1 a Study the limit of f in + ∞ . b Show that the straight line Δ of equation y =2 x − 2 is asymptote to C . c Investigate the relative position of C and Δ . 2 a Calculate f ( x ) and show that : f ( x ) = x e − x + 2 · 1 − e − x b Deduce that, for any strictly positive real x , f ( x ) > 0 . c Specify the value of f (0) , then establish the table of variations of f . 3 Using integration by parts, calculate the area, expressed in cm 2 , of the plane region bounded by the curve C , the line Δ and the lines of equations x =1 and x =3 . 4 a Determine the point A of C where the tangent to C is parallel to Δ . b Calculate the distance, expressed in cm , from point A to line Δ . https://chingmath.fr chapExoCorrec/3211 sacados/3211 Antilles-Guyane Septembre 2005 4 points -12345I-0.6-0.5-0.4-0.3-0.2-0.100.10.20.30.40.50.60.70.8 chapExoCorrec/3220 sacados/3220 France Septembre 2004 4 points chapExoCorrec/3236 sacados/3236
-123I-1234JO 234I234JO E.3249 The exercise includes an appendix to be submitted with the copy. Consider the functions f and g defined on the interval 0 ; + ∞ , by: f ( x ) = ln( x +1) ; g ( x ) = e x − 1 We denote by C f and C g the representative curves of the functions f and g in an orthonormal coordinate system O ; −→ i ; −→ j . These curves are plotted on the attached sheet, which the candidate may use as they see fit ; this attachment should be included with the copy, along with any additions made by the candidate. 1 Check that the curves C f and C g have a common tan-gent at point O (0 ; 0) . Specify the position of the curve C f relative to this tangent. 2 Show that the curves C f and C g are symmetrical with respect to the line with equation y = x . 3 Let a be a strictly positive real number. We propose to calculate the number I ( a ) = a 0 ln( x +1) d x . a Using area considerations, demonstrate that : I ( a ) = a · ln( a +1) − ln( a +1) 0 e x − 1 d x b Deduce the value of I ( a ) . c Find the value of I ( a ) by performing integration by parts. https://chingmath.fr -123I-1234JO chapExoCorrec/3249 sacados/3249 234I234JO
E.3272 Purpose of the exercise: ap-proximate ln(1+ a ) by a polynomial of degree 5 when a be-longs to the interval [0 ; + ∞ [ . Let a ∈ [0 ; + ∞ [ . We note I 0 ( a )= a 0 1 1+ t d t and for k ∈ N ∗ , we set : I k ( a ) = a 0 ( t − a ) k (1 + t ) k +1 d t 1 Calculate I 0 ( a ) as a function of a . 2 Using integration by parts, express I 1 ( a ) in terms of a . 3 Using integration by parts, demonstrate that : I k +1 ( a ) = ( − 1) k +1 a k +1 k + 1 + I k ( a ) for all k ∈ N ∗ 4 Let P be the polynomial defined on R : P ( x ) = 1 5 x 5 − 1 4 x 4 + 1 2 x 3 − 1 2 x 2 + x . Prove by calculating I 2 ( a ) , I 3 ( a ) and I 4 ( a ) , that : I 5 ( a ) = ln(1+ a ) − P ( a ) 5 Let J ( a )= a 0 ( t − a ) 5 d t . Calculate J ( a ) . 6 a Prove that for all t ∈ [0 ; a ] : ( t − a ) 5 (1 + t ) 6 ( t − a ) 5 b Show that for all a ∈ [0 ; + ∞ [ : J ( a ) I 5 ( a ) 0 7 Deduce that for any a ∈ [0 ; + ∞ [ : ⏐ ⏐ ln(1+ a ) − P ( a ) ⏐ ⏐ a 6 6 . 8 Justifying your answer, determine an interval over which P ( a ) is an approximate value of ln(1+ a ) to within 10 − 3 . E.4000 Let f be the function defined on the interval 0 ; + ∞ par : f ( x ) = x − 1 2 − e − x Its representative curve C is plotted in the orthonormal coor-dinate system below (graphical unit 2 cm ) . 1 a Study the limit of f at + ∞ . b Show that the line Δ with equation y =2 x − 2 is an asymptote to C . c Study the relative position of C and Δ . 2 a Calculate f ( x ) and show that : f ( x ) = x · e − x + 2 · 1 − e − x . b Deduce that, for any strictly positive real number x : f ( x ) > 0 c Specify the value of f (0) , then establish the table of variations of f . 3 Using integration by parts, calculate the area, expressed in cm 2 , of the plane region bounded by the curve C , the line Δ and the lines with equations x =1 and x =3 . 4 a Determine the point A of C where the tangent to C is parallel to Δ . b Calculate the distance, expressed in cm , from point A to line Δ . E.4128 The aim of this exercise is to de-termine an approximate value to 10 − 2 of the integral: I = 1 0 e − x 2 − x d x 1 a Study the variations of the function : f : x ↦−→ f ( x ) = e − x 2 − x sur l’intervalle 0 ; 1 . b Show that, for any real x on the interval 0 ; 1 , we have : 1 e f ( x ) 1 2 2 Let J and K be the integrals defined by: J = 1 0 2 + x · e − x d x ; K = 1 0 x 2 · f ( x ) d x a By means of integration by parts, prove that : J = 3 − 4 e b Use a frame of f ( x ) obtained previously to show that : 1 3 · e K 1 6 . c Demonstrate that : J + K =4 · I . d Deduce from all the above a frame for I , then give an approximate value to 10 − 2 nearest I . 7. Integral by parts - sequences and annals E.3196 1 Let f be the function defined on R by: f ( x ) = x 2 e 1 − x We denote by C its representative curve in an orthonor-mal coordinate system O ; −→ i ; −→ j with unit length 2 cm . a Determine the limits of f at −∞ and + ∞ ; what graph-ical consequence for C can be drawn from this? b Justify that f is differentiable on R . Determine its derivative function f . c Draw up the table of variations of f and plot the curve C . 2 Let n be a non-zero natural number. Consider the inte-gral I n defined by: I n = 1 0 x n e 1 − x d x a Establish a relationship between I n +1 and I n . b Calculate I 1 , then I 2 . c Give a graphical interpretation of the number I 2 . This will be shown on the graph in question 1 c . 3 a Demonstrate that for any real number x of [0 ; 1] and for any natural number n not equal to zero, the following inequality holds : x n x n e 1 − x x n e b Deduce a bound for I n and then the limit of I n when n tends towards + ∞ . https://chingmath.fr chapExoCorrec/3272 sacados/3272 Antilles-Guyane Juin 2004 7 points chapExoCorrec/4000 sacados/4000 Amerique du Nord Juin 2005 5 points chapExoCorrec/4128 sacados/4128 chapExoCorrec/3196 sacados/3196 France Juin 2006 5 points
E.3205 Part A : study of a function Let f be the function defined on the interval 0 ; + ∞ par f ( x ) = x ln( x + 1) Its representative curve ( C ) in an orthogonal reference frame O ; −→ u ; −→ v est given in the appendix. 1 a Montrer that the function f est strictly increasing on the interval 0 ; + ∞ . b Is the x-axis tangent to the curve ( C ) au point O ? 2 On pose : I = 1 0 x 2 x + 1 d x a Determine three real a , b et c tels that, for any x = 1 , x 2 x + 1 = ax + b + c x + 1 b Calculate I . 3 Using integration by parts and the result obtained in question 2 , calculate, in area units, the area A of the part of the plane bounded by the curve ( C ) et the straight lines of equations x =0 , x =1 et y =0 . 4 Show that the equation f ( x )=0.25 admits a single solu-tion on the interval [0 ; 1] . Note ¸ this solution. Give a frame for ¸ of amplitude 10 − 2 . Part B: studying a sequence The sequence ( u n ) is defined on N by: u n = 1 0 x n ln( x + 1) d x 1 Determine the direction of variation of the sequence ( u n ) . Will the sequence ( u n ) converge? 2 Prove that for any natural number n not equal to zero: 0 u n ln 2 n + 1 . Deduce the limit of the sequence ( u n ) . E.3239 In this exercise we are interested in a sequence of rational numbers that converges to e 2 . We define, for any natural number n 1 , the integral: I n = 2 0 1 n ! (2 − x ) n e x d x 1 Calculate I 1 . 2 Establish that for any natural number n 1 : 0 I n 2 n n ! e 2 − 1 3 Using integration by parts, show that for any natural number n 1 : I n +1 = I n − 2 n +1 ( n + 1)! 4 Demonstrate by recurrence that : e 2 = 1 + 2 1! + 2 2 2! + · · · + 2 n n ! + I n 5 For any natural number n 1 , u n = 2 n n ! . a Calculate u n +1 u n and prove that for any natural number n 3 : u n +1 1 2 u n . b Deduce that for any natural number n 3 : 0 u n u 3 · 1 2 n − 3 6 Deduce the limit of the sequence ( u n ) and then that of the sequence ( I n ) . 7 Finally justify that : e 2 = lim n ↦→ + ∞ 1 + 2 1! + 2 2 2! + · · · + 2 n n ! https://chingmath.fr chapExoCorrec/3205 sacados/3205 Liban Mai 2006 7 points chapExoCorrec/3239 sacados/3239 Asie Juin 2005 7 points
E.3988 Let f be a function defined for all real numbers x by: f ( x ) = (1 + x ) · e − x The plane is referenced to an orthonormal coordinate system O ; −→ i ; −→ j with unit length 1 cm . 1 a Study the sign of f ( x ) on R . b Determine the limit of the function f at −∞ . Determine the limit of the function f at + ∞ . c We denote f as the derivative of the function f on R . Calculate, for any real number x , f ( x ) . Deduce the variations of the function f on R . d Plot the graph of the function f on the interval − 2 ; 5 . 2 We note I n the sequence defined for any natural num-ber n by: I n = n − 1 f ( x ) d x In this question, we will not attempt to calculate the exact value of I n as a function of n . a Show that, for all n ∈ N : I n 0 . b Show that the sequence I n is increasing. 3 a Using integration by parts, show that, for all real numbers a and b : b a f ( x ) d x = − 2 − b · e − b + 2 + a · e − a b Deduce the expression of I n as a function of n . c Determine : lim n ↦→ + ∞ I n . d Provide a graphical interpretation of this limit. 4 Determine a ∈ R such that : a − 1 f ( x ) d x = e Does this integral calculation correspond to an area cal-culation? E.3999 Consider the sequences x n and y n defined for any non-zero natural number n by: x n = 1 0 t n · cos t d t ; y n = 1 0 t n · sin t d t 1 a Show that the sequence x n has positive terms. b Study the variations of the sequence x n . c What can we deduce about the convergence of the se-quence x n ? 2 a Show that, for any non-zero natural number n : x n 1 n + 1 b Deduce the limit of the sequence x n . 3 a Using integration by parts, show that, for any non-zero natural number n : x n +1 = − ( n + 1) · y n + sin(1) . b Deduce that : lim n ↦→ + ∞ y n = 0 4 We admit that, for any non-zero natural number n : y n +1 = ( n + 1) · x n − cos(1) . Determine : lim n ↦→ + ∞ n · x n and lim n ↦→ + ∞ n · y n 8. Integral by parts - differential equation and annals E.3125 Part A 1 Solve the differential equation : y  − 4 y + 4 y = 0 2 Determine the solution ffi of this equation, defined on R and which checks the conditions : ffi (0) = 0 ; ffi (0) = − e Part B 1 Consider the function f defined on R by: f ( x ) = − x · e 2 x +1 a What is, depending on the values of x , the sign of f ( x ) ? b Investigate the direction of variation of f . c Determine the limits of f in + ∞ and in −∞ . d Draw up the table of variations of f . e We call ( C ) the graphical representation of f in an or-thonormal frame O ; −→ i ; −→ j (unité graphique: 4 cm ) What is the tangent to ( C ) at point O ? Write an equation of the tangent T to ( C ) at the point of abscissa ( − A ) . f We call (Γ) the graphical representation in the refer-ence frame O ; −→ i ; −→ j of the function g defined on R by: g ( x ) = e x What is the tangent to (Γ) at the point of abscissa ( − A ) ? 2 We call h the function defined on R by: h ( x ) = 1 + e · x · e x a Study the direction of variation of h . Deduce the sign of h ( x ) depending on the values of x . b Investigate the position of ( C ) relative to (Γ) . c Plot, on the same graph, the curves T , ( C ) and (Γ) . 3 Let m be any real and M the point on the curve (Γ) of abscissa M . a Write an equation for the tangent D to (Γ) at M . b The tangent D intersects the coordinate axes at A and B . Calculate, as a function of m ; the coordinates of the middle J of the segment [ AB ] . c Prove that J belongs to ( C ) . d Trace ( D ) and J to m =0 . Part C 1 Let x be any real. Using integration by parts, calculate the integral: I ( x ) = x 0 t · e 2 t d t 2 Let x be a negative real. https://chingmath.fr chapExoCorrec/3988 sacados/3988 chapExoCorrec/3999 sacados/3999 sacados/3125 France Septembre 1998 11 points
x-4-3-2-101234y-2-1123 Calculate the area A ( x ) , expressed in cm 2 , of the set of points N whose coordinates ( u ; v ) verify: x u 0 0 v f ( x ) 3 Calculate A ( − 1) . does 4 A ( x ) admit a limit when x tends to minus infinity? If so, which one? E.3189 Part A Consider the differential equation : ( E ) : y + y = e − x 1 Show that the function u defined on the set R of real numbers by: u ( x ) = x e − x is a solution of ( E ) . 2 Solve the differential equation : ( E 0 ) : y + y = 0 3 Show that a function v , defined and derivable on R , is a solution of ( E ) if, and only if, v − u is a solution of ( E 0 ) . 4 Deduce all solutions of ( E ) . 5 Determine the function f 2 , solution of ( E ) , which takes the value 2 in 0. Part B k being a given real number, let f k be the function defined on the set R by: f k ( x ) = ( x + k ) · e − x We denote C k the representative curve of the function f k in an orthonormal frame O ; −→ i ; −→ j . 1 Determine the limits f k in −∞ and + ∞ . 2 Calculate f k ( x ) for any real x . 3 Deduce the table of variations of f k . Part C 1 Consider the sequence of integrals ( I n ) defined by: I 0 = 0 − 2 e − x d x and for any natural number n 1 by: I n = 0 − 2 x n e − x d x a Calculate the exact value of the integral I 0 . b Using integration by parts, demonstrate equality: I n +1 = ( − 2) n +1 · e 2 + ( n + 1) · I n c Deduce the exact values of the integrals I 1 and I 2 . 2 The graph below represents a curve C k which is the graphical representation of a function f k defined at B . a Using the information given by the graph, determine the value of the corresponding real number k . b Let S be the area of the hatched part (in area units) ; express S as a function of I 1 and I 0 and deduce its exact value. E.3214 Part A The function f is defined on the interval 0 ; + ∞ by: f ( x ) = 20 x + 10 e 1 2 x We denote C as the curve representing the function f in an or-thonormal coordinate system O ; −→ i ; −→ j (graphic unit 1 cm ) . 1 Study the limit of the function f at + ∞ . 2 Study the variations of the function f and draw up its variation table. 3 Establish that the equation f ( x )=10 has a unique strictly positive solution ¸ in the intervalle 0 ; + ∞ . Give the rounded value of ¸ to 10 − 3 near. 4 Plot the curve C . 5 Calculate the integral: I = 3 0 f ( x ) d x Part B Let y ( t ) be the value, in degrees Celsius, of the temperature of a chemical reaction at time t , where t is expressed in hours. The initial value, at time t =0 , is y (0)=10 . We assume that the function which, for any real number t belonging to the interval 0 ; + ∞ is associated with y ( t ) , is the solution to the differential equation : ( E ) : y + 1 2 · y = 20 · e − 1 2 t 1 Check that the function f studied in section A is a so-lution to the differential equation ( E ) over the interval 0 ; + ∞ . 2 We propose to show that this function f is the unique solution of the differential equation ( E ) , defined on the interval 0 ; + ∞ , which takes the value 10 at time 0 . a Let g be any solution of the differential equation ( E ) , defined on 0 ; + ∞ verifying g (0)=10 . Show that the function g − f is a solution, on the interval 0 ; + ∞ , of the differential equation : ( E ) : y + 1 2 · y = 0 b Solve the differential equation ( E ) . c Conclude. 3 After how long does the temperature of this chemical re-action drop back to its initial value? The result should be rounded to the nearest minute. 4 The value „ in degrees Celsius of the average temperature at this chemical reaction during the first three hours is the average value of the function f over the intervalle https://chingmath.fr chapExoCorrec/3189 sacados/3189 Asie Juin 2006 7 points x-4-3-2-101234y-2-1123 chapExoCorrec/3214 sacados/3214 France Septembre 2005 7 points
-4-3-2-1I-1JO 0 ; 3 . Calculate the exact value of „ , then give its value rounded to the nearest degree. 9. Complete and volumes E.4018 We denote by f the function de-fined on the set R of real numbers by: f ( x ) = 1 1 + e − x Let – be a positive real, note V ( – ) the integral: 0 − λ ı · f ( x ) 2 d x It is accepted that V ( – ) is a measure, expressed in units of vol-ume, of the volume generated by rotation about the abscissa axis, of the portion of the curve C obtained for − – x 0 . 1 Determine the real numbers a and b such that for any real number x : e 2 · x e x + 1 2 = a · e x e x + 1 + b · e x e x + 1 2 2 Express V ( – ) as a function of – . 3 Determine the limit of V ( – ) when – tends to + ∞ . E.4328 Consider the function f defined on 0 ; + ∞ by: f ( x ) = x 2 · ln x The curve ( C ) is the representative curve of the function f in the plane provided with an orthonormal reference frame O ; −→ i ; −→ j . Consider the solid obtained by rotation about the axis ( Ox ) of the planar region bounded by the curve ( C ) , the axis ( Ox ) and the straight lines of equations : x = 1 e ; x = 1 We denote V a measure, expressed in units of volume, of the volume of this solid and admit that : V = 1 1 e ı · f ( x ) 2 d x 1 Show that a primitive of the function : x ↦−→ x 4 · ln x on 0 ; + ∞ is the function : x ↦−→ x 5 25 · 5 · ln x − 1 . 2 Deduce, using integration by parts, that : V = ı 125 · 2 − 37 e 5 https://chingmath.fr chapExoCorrec/4018 sacados/4018 -4-3-2-1I-1JO sacados/4328 Extrait de Polynesie Juin 2011
234I23JOAB -2-12I23JO E.3216 Let f be the function defined on 0 ; + ∞ by: f ( x )= x · e − x +2 The two parts can be approached independently. Part A 1 Draw up a table of variations of f over [0 ; + ∞ [ and de-termine any asymptotes of the representative curve. 2 a Plot the curves of the function f and the natural logarithm function on the graphing calculator; note L the latter. Use this graph to conjecture the number of solutions to the equation f ( x ) = ln( x ) sur 1 ; + ∞ . b Show that the function g defined on R ∗ + by: g ( x ) = ln( x ) − f ( x ) is strictly increasing on 1 ; + ∞ . Deduce that the equation f ( x )=ln( x ) has a unique solution ¸ on 1 ; + ∞ [ . c Determine the value of ¸ rounded to 10 − 3 close. Part B 1 Using double integration by parts, determine : I = 3 0 x 2 e − 2 x d x 2 We define the solid S obtained by revolution around the axis ( Ox ) of the curve with equation y = f ( x ) for 0 x 3 in the plane ( xOy ) (orthonormal unit reference frame 4 cm ) . Recall that the volume V of the solid is given by: V = ı 3 0 f ( x ) 2 d x a Express V in terms of I . b Determine the volume of the solid rounded to 1 cm 3 . E.3229 Shown above, in an orthonormal O ; −→ i ; −→ j , the representa-tive curve of the function f derivable on R , solution of the differential equation : ( E ) : y + y = 0 and such that f (0)= e . 1 Determine f ( x ) for any real x . 2 Let t be a given real from the interval 1 ; e . Solve in R the equation e 1 − x = t of unknown x . 3 Let A be the point of abscissa 0 and B the point of ab-scissa 1 of the curve. Consider the solid obtained by rotation around the y-axis of the curve arc AB as shown below. Note V its volume. Assume that : V = ı · e 1 (1 − ln t ) 2 d t Calculate V using two successive integrations by parts. 10. Former annuals (before 2012) E.3138 1 Let f be the function defined on R by: f ( x ) = 2 x 3 − 4 x 2 · e − x a Determine the limits of f in −∞ and + ∞ . b Calculate f ( x ) and show that : f ( x ) = 2 x − x 2 + 5 x − 4 · e − x . c Draw up the table of variations of f . d Draw the curve C representative of f in an orthonor-mal frame O ; −→ i ; −→ j (unité graphique: 1 cm ) . 2 For n ∈ N ∗ , we pose : I n = 1 0 x n · e − x dx a Using integration by parts, calculate I 1 . https://chingmath.fr chapExoCorrec/3216 sacados/3216 Antilles-Guyane Septembre 2004 5 points chapExoCorrec/3229 sacados/3229 234I23JOAB -2-12I23JO chapExoCorrec/3138 sacados/3138
MNI2JOPartieBPartieA b We admit that, for any n greater than or equal to 2 : I n = n · I n − 1 − 1 e . Determine I 2 and I 3 . c Let A be the area, expressed in cm 2 , of the domain bounded by the x-axis, the curve ( C ) and the straight lines of equation x =0 and x =1 . Calculer A . 3 Let u be a function defined and derivable on R . We define the function v on 0 ; + ∞ by v ( x )= u 1 x . a It is assumed that u is increasing on the interval [ a ; b ] (where 0 <a<b ) . Determine the direction of variation of v on 1 b ; 1 a . b We now define the function g by g ( x )= f 1 x on 0 ; + ∞ , where f is the function defined in question 1 . Determine the limits of g in 0 and in + ∞ . c Deduce from the previous questions the table of varia-tions of the function g on the interval 0 ; + ∞ . E.3201 Part One Calculate the integral: 1 0 x · e x d x . Part Two The figure below shows a rectan-gular target OIMN such that, in the orthonormal coordinate system O ; −→ OI ; −→ OJ , the curved line C con-necting point O to point M is part of the curve representing the function f defined on R by f ( x )= x · e x . This curve divides the target OIMN into two parts, A and B , as shown in the figure below. One game involves throwing a dart that hits either the outside of the tar-get or one of the sections A or B . It is assumed that the dart cannot hit any of the target’s borders or the curve C . A statistical study has shown that the dart falls outside the target with a probability of 1 2 and that the probabilities of hitting the parts A and B are proportional to their respective areas. 1 Show that the probability of reaching part A is equal to 1 2e . What is the probability of reaching part B ? 2 Three darts are thrown independently: a Let X be the random variable that equals the number of darts that hit the game A . Define the probability distribution of X . Deduce the exact value of its math-ematical expectation. b Let E be the event : ˇ Exactly two darts hit the A ı part. Calculate the probability of E rounded to the nearest thousandth. c Let F be the event : ˇ all three darts hit the part B ı. Calculate the probability of F (exact value will be given) . Knowing that none of the darts hit the outside of the target, what is the probability that all three will be in the B part? 3 This time we independently throw n darts. a Determine as a function of n the probability p n that at least one of the darts hits the A part. b Determine the smallest integer n ∈ N such that p n 0.99 . E.3222 The curve C given below is the graphical representation of the function f defined on ]0 ; + ∞ [ by: f ( x ) = ln x x + 1 − x 1 a Show that f is differentiable and that, for any strictly positive x , f ( x ) has the sign of : N ( x ) = − 2 x · x − 1 + ln x b Calculate N (1) and determine the sign of N ( x ) by dis-tinguishing between the cases 0 <x< 1 and x> 1 . c Deduce the direction of variation of f on 0 ; + ∞ and the coordinates of the point with the maximum ordi-nate C . 2 Note A ( ¸ ) the area, expressed in area units, of the part of the plane shaded in gray on the figure, where ¸ denotes a real of 0 ; 1 . a Express A ( ¸ ) as a function of ¸ (integration by parts may be used) . b Calculate the limit of A ( ¸ ) when ¸ tends to 0. Give a graphical interpretation of this limit. We define a sequence u n n ∈ N by its first term u 0 element of 1 ; 2 and : for any natural number n : u n +1 = ln u n u n + 1 3 a Demonstrate, for any real x element of [1 ; 2] , the double inequality: 0 ln x x 1 b Demonstrate by recurrence that, for any natural num-ber n , u n belongs to 1 ; 2 . 4 Noting that, for any n ∈ N , u n +1 = f ( u n )+ u n , determine the direction of variation of the sequence u n . 5 a Show that the sequence ( u n ) n ∈ N is convergent. Let ‘ be its limit. b Determine the exact value of ‘ . https://chingmath.fr chapExoCorrec/3201 sacados/3201 MNI2JOPartieBPartieA chapExoCorrec/3222 sacados/3222
¸23I-4-3-2-1JO E.3262 Consider the function f defined on the interval 0 ; + ∞ by: f (0) = 1 f ( x ) = 1 2 · x 2 · 3 − 2 · ln x + 1 if x> 0 We note C the representative curve of f in an orthonormal coordinate system O ; −→ i ; −→ j . Part A 1 a Calculate lim x ↦→ 0 f ( x ) . What can we deduce about the function f ? b Determine the limit of f in + ∞ . 2 a Study the derivability of f in 0 . b Show that f is derivable on the interval 0 ; + ∞ and calculate f ( x ) for x> 0 , f denoting the function de-rived from f . 3 Study the direction of variation of f on [0 ; + ∞ [ , then draw up its table of variations. 4 Show that the equation f ( x )=0 has a unique solution ¸ on the interval 0 ; + ∞ . Determine the value of ¸ rounded to the nearest 10 − 2 . Part B 1 Calculate an equation of the tangent D to the curve C at the point with abscissa x =1 . 2 Consider the function g : x ↦→ f ( x ) − 2 x − 1 2 defined on the interval 0 ; + ∞ . a Calculate g ( x ) , then g  ( x ) where g and g  denote the first and second derivative functions of g , respectively. Study the direction of variation of g . Deduce the sign of g ( x ) on 0 ; + ∞ . b Study the direction of variations of g . Deduce the position of the curve C relative to the tan-gent D . 3 Construct the curve C and the tangent D (graphic unit 2 cm ) Part C n is a non-zero natural number. 1 Express in terms of n the real number: I n = 1 1 n x 2 · ln x d x (we can use integration by parts) 2 Deduce, based on the integer n , the area A n expressed in cm 2 of the plane domain bounded limited by the curve C , the tangent D and the two lines of equations x = 1 n and x =1 . 3 calculate lim n ↦→ + ∞ A n and interpret the result obtained. E.3251 Part A Consider the sequence ( u n ) defined, for all natural numbers n not equal to zero, by: u n = 1 0 (1 − t ) n · e t d t . 1 Show that the function f : t ↦→ (2 − t ) · e t is a primitive of g : (1 − t ) · e t sur 0 ; 1 . Deduce the value of u 1 . 2 Show using integration by parts that, for any n non-zero : u n +1 = ( n + 1) · u n − 1 ( R ) Part B First, we look at what two different calculators display for the approximate values of the first 25 terms of the sequence ( u n ) using the recurrence relation ( R ) above. Here are the results displayed by these two calculators : Value of n Value of u n displayed by the first calculator the second calculator 1 7.1828182845E-01 7.1828182846E-01 2 4.3656365691E-01 4.3656365692E-01 3 3.0969097075E-01 3.0969097076E-01 4 2.3876388301E-01 2.3876388304E-01 5 1.9381941508E-01 1.9381941520E-01 6 1.6291649051E-01 1.6291649120E-01 7 1.4041543358E-01 1.4041543840E-01 8 1.2332346869E-01 1.2332350720E-01 9 1.0991121828E-01 1.0991156480E-01 10 9.9112182825E-02 9.9115648000E-01 11 9.0234011080E-02 9.0272128000E-02 12 8.2808132963E-02 8.3265536000E-02 13 7.6505728522E-02 8.2451968000E-02 14 7.1080199309E-02 1.5432755200E-01 15 6.6202989636E-02 1.3149132800E+00 16 5.9247834186E-02 2.0038612480E+01 17 7.2131811612E-03 3.3965641216E+02 18 -8.7016273909E-01 6.1128154189E+03 19 -1.7533092042E+01 1.1614249296E+05 20 -3.5166184085E+02 2.3228488592E+06 21 -7.3858986580E+03 4.8779825043E+07 22 -1.6249077047E+05 1.0731561499E+09 23 -3.7372887209E+06 2.4682591448E+10 24 -8.9694930302E+07 5.9238219474E+11 25 -2.2423732585E+09 1.4809554869E+13 What conjecture can be made about the convergence of the https://chingmath.fr ¸23I-4-3-2-1JO chapExoCorrec/3262 sacados/3262 chapExoCorrec/3251 sacados/3251 Liban juin 2005 8 points
-1012345678-2-1123ABC sequence ( u n ) when examining the results obtained with the first calculator? And with the results obtained with the sec-ond calculator? Part C In this part, we propose to study the sequence ( u n ) from the definition : for any n ∈ N ∗ : u n = 1 0 (1 − t ) n · e t d t 1 Show that for any n ∈ N ∗ : u n 0 . 2 a Monstrate that for any real t of the interval 0 ; 1 et for any n ∈ N ∗ : (1 − t ) n e t e × (1 − t ) n b Deduce that for any n non-null: u n e n +1 . 3 Determine the limit of the sequence ( u n ) . Partie D In this part, we propose to exploit the recurrence relation ( R ) verified by the sequence ( u n ) : u n +1 = ( n + 1) · u n − 1 Given a real a , consider the sequence ( v n ) defined by: v 1 = a ; v n +1 =( n +1) · v n − 1 pour all n ∈ N ∗ 1 Using reasoning by recurrence, show that for any non-zero natural number n : v n = u n + ( n !) · ( a + 2 − e ) where n ! désigne the product of n premiers non-zero nat-ural numbers. 2 Study the behavior of the sequence ( v n ) to infinity de-pending on the values of a . (Recall that lim n ↦→ + ∞ n !=+ ∞ ) 3 Indeduce a reason that could explain the results dis-played by the two calculators. 11. Unclassified financial years E.79 Part I Let f be a function defined on the interval 1 ; 8 , strictly de-creasing, whose graphical representation C in an orthonormal reference frame is given opposite. The curve C contains the points A (1 ; 2) , B (2 ; 0) and C (4 ; − 1) . 1 Using the graphical representation, give, according to the values of x , the sign of f ( x ) . 2 It is assumed that, for any x in the interval [1 ; 8] f ( x ) is written : f ( x )= − 2+ 4 x Find by calculation, the result of 1 . Part II Consider the function F defined on the interval 1 ; 8 by: F ( x ) = 5 − 2 x + 4 ln( x ) . 1 Show that F has as derivative the function f of the part I . 2 Study the variations of the function F on the interval 1 ; 8 , then draw up its table of variation. 3 C F denotes the representative curve of the function F in an orthogonal reference frame with graph units : abscissa 1 cm , ordinate 2 cm . a Consider the straight line Δ , tangent to the curve C F at its point of abscissa 1. Show that the directing co-efficient of the line Δ is equal to 2. b Draw the curve C F and the straight line Δ . Form : The derivative of the function ln on the interval 0 ; + ∞ is the function which, at x , associates 1 x . E.4014 Organized restitution of knowl-edge: Demonstrate the formula for integration by parts using the formula for the derivation of a product of two derivable func-tions, continuously derived over an interval a ; b . Let the two integrals defined by: I = π 0 e x · sin x d x ; J = π 0 e x · cos x d x 1 Demonstrate that : I = − J ; I = J +e π +1 2 Deduce the exact values of I and J . E.3996 Let I be an interval of R . So let u and v be two continuous functions, derivable on I such that u and v are continuous on I . Recall and demonstrate the formula for integration by parts over an interval a ; b of I . https://chingmath.fr chapExoCorrec/79 sacados/79 -1012345678-2-1123ABC chapExoCorrec/4014 sacados/4014 chapExoCorrec/3996 sacados/3996 Extrait d'Antilles Guyane Septembre 2007