Calcul intégral

QCM Bilan Numéro 1

Exercice 1

Cet exercice est un questionnaire à choix multiples (QCM). Pour chacune des questions ci-dessous, une seule des réponses est exacte. Pour chaque question, vous devez bien sur justifier.
1

D=10(3t45t3+2t26t+2)dtD=\int _{-1}^{0}\left(3t^{4} -5t^{3} +2t^{2} -6t+2\right) dt est égale à :
a.\bf{a.} 44960\frac{449}{60}                                                                                               \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} 45160\frac{451}{60}

c.\bf{c.} 45360\frac{453}{60}                                                                                              \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 45760\frac{457}{60}

Correction
2

A=013xe3x2dxA=\int _{0}^{1}3xe^{3x^{2} } dx est égale à :
a.\bf{a.} 12(e31)\frac{1}{2} \left(e^{3} -1\right)                                                                                                   \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} e31e^{3} -1

c.\bf{c.} 2(e31)2 \left(e^{3} -1\right)                                                                                                    \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 12(e3+1)\frac{1}{2} \left(e^{3} +1\right)

Correction
3

C=123x25x+94xdxC=\int _{1}^{2}\frac{3x^{2} -5x+9}{4x} dx est égale à :
a.\bf{a.} 158+94ln(2)\frac{15}{8} +\frac{9}{4} \ln \left(2\right)                                                                                               \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} 158+94ln(2)-\frac{15}{8} +\frac{9}{4} \ln \left(2\right)

c.\bf{c.} 18+94ln(2)\frac{1}{8} +\frac{9}{4} \ln \left(2\right)                                                                                                  \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 18+94ln(2)-\frac{1}{8} +\frac{9}{4} \ln \left(2\right)

Correction
4

B=12(5x47x5)dxB=\int _{1}^{2}\left(\frac{5}{x^{4} } -\frac{7}{x^{5} } \right) dx est égale à :
a.\bf{a.} 31192-\frac{31}{192}                                                                                               \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} 35192\frac{35}{192}

c.\bf{c.} 33192-\frac{33}{192}                                                                                              \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 35192-\frac{35}{192}

Correction
5

Soit xx un réel appartenant à l'intervalle [1;5]\left[1;5\right]. Si pour tout x[1;5]x\in \left[1;5\right], on a : f(x)g(x)f\left(x\right)\ge g\left(x\right) alors :
a.\bf{a.} 15f(x)dx15g(x)dx\int _{1}^{5}f\left(x\right) dx\ge \int _{1}^{5}g\left(x\right) dx                                                                                               \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} 15g(x)dx15f(x)dx\int _{1}^{5}g\left(x\right) dx\ge \int _{1}^{5}f\left(x\right) dx

c.\bf{c.} 51f(x)dx15g(x)dx\int _{5}^{1}f\left(x\right) dx\ge \int _{1}^{5}g\left(x\right) dx                                                                                              \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} aucune des trois propositions proposées ne convient.

Correction
6

174x8dx\int _{1}^{7}\left|4x-8\right| dx est égale à :
a.\bf{a.} 5252                                                                                               \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} 2525

c.\bf{c.} 52-52                                                                                            \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 25-25

Correction
7

On considère la fonction ff continue sur [3;5]\left[3;5 \right] définie par f(x)=4x1f\left(x\right)=4x-1. La valeur moyenne de ff sur l'intervalle [3;5]\left[3;5\right] est égale à :
a.\bf{a.} 1515                                                                                             \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} 2020

c.\bf{c.} 3030                                                                                            \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 4040

Correction
8

G=1e(x+3)ln(x)dxG=\int _{1}^{e}\left(-x+3\right)\ln \left(x\right)dx est égale à :
a.\bf{a.} 14e2+114\frac{1}{4} e^{2} +\frac{11}{4}                                                                                             \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} 14e2114-\frac{1}{4} e^{2} -\frac{11}{4}

c.\bf{c.} 14e2114\frac{1}{4} e^{2} -\frac{11}{4}                                                                                             \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 14e2+114-\frac{1}{4} e^{2} +\frac{11}{4}

Correction
9

Soit nn un entier naturel tel que : In=01xnexdxI_{n} =\int _{0}^{1}x^{n} e^{x} dx . A l'aide d'une intégration par parties, on peut affirmer que : In+1=I_{n+1}=
a.\bf{a.} In+1=enInI_{n+1}=e-nI_{n}                                                                                       \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} In+1=e+(n+1)InI_{n+1}=e+\left(n+1\right)I_{n}

c.\bf{c.} In+1=eInI_{n+1}=e-I_{n}                                                                                             \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} In+1=e(n+1)InI_{n+1}=e-\left(n+1\right)I_{n}

Correction
10

H=0π(12cos(2x)(4+sin(2x))5)dxH=\int _{0}^{\pi }\left(12\cos \left(2x\right)\left(4+\sin \left(2x\right)\right)^{5} \right)dx est égale à :
a.\bf{a.} 00                                                                                       \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} 1-1

c.\bf{c.} 11                                                                                        \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 2×462\times4^{6}

Correction
11

J=103ex1+4exJ=\int _{-1}^{0}\frac{3e^{-x} }{1+4e^{-x} } est égale à :
a.\bf{a.} 14(ln(1+4e5))\frac{1}{4} \left(\ln \left(\frac{1+4e}{5} \right)\right)                                                                                       \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; b.\bf{b.} 34(ln(1+4e5))\frac{3}{4} \left(\ln \left(\frac{1+4e}{5} \right)\right)

c.\bf{c.} 54(ln(1+4e5))\frac{5}{4} \left(\ln \left(\frac{1+4e}{5} \right)\right)                                                                                        \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; d.\bf{d.} 74(ln(1+4e5))\frac{7}{4} \left(\ln \left(\frac{1+4e}{5} \right)\right)

Correction
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