\documentclass[11pt,a4]{article} \usepackage{amsfonts} \usepackage{stmaryrd} \usepackage[french]{babel} \usepackage[applemac]{inputenc} \usepackage{a4} \setlength{\oddsidemargin}{-0mm} \setlength{\textwidth}{158mm} \setlength{\textheight}{230mm} \setlength{\topmargin}{0mm} \setlength{\headsep}{-2mm} \renewcommand{\baselinestretch}{1.2} \pagestyle{empty} \newtheorem{exo}{Exercice} \newtheorem{correction}{Correction} \begin{document} \def\Argch{\mathop{\rm Argch}} \def\Argsh{\mathop{\rm Argsh}} \def\Argth{\mathop{\rm Argth}} \def\Arccos{\mathop{\rm Arccos}} \def\Arcsin{\mathop{\rm Arcsin}} \def\Arctan{\mathop{\rm Arctan}} \def\ch{\mathop{\rm ch}} \def\sh{\mathop{\rm sh}} \def\th{\mathop{\rm th}} \def\tanh{\mathop{\rm th}} \def\N{\mathop{\mathbb N}} \def\Z{\mathop{\mathbb Z}} \def\R{\mathop{\mathbb R}} \def\donne{\Rightarrow} \def\Sup{\mathop{\rm Sup}} \def\C{\mathop{\mathbb C}} \noindent{\large\bf Universit\'e Henri Poincar\'e\hfill Facult\'e des Sciences et Technologies} \noindent{\sl {Licences Math\'ematiques, SPI et Informatique \hfill Automne 2010}} \vspace{0.5cm} \begin{center} {\Large Liste d'exercices n$^{\circ}$5: Calcul de primitives } \end{center} \vskip0,5cm \vskip1cm \section{Fractions rationnelles} \begin{exo} Calculer les primitives suivantes : $$\int{5x-12\over {x(x-4)}}\ dx\ ,\ \int{37-11x\over {(x+1)(x-2)(x-3)}}\ dx\ ,\ \int{6x-11\over {(x-1)^2}}\ dx$$ $$ \int \frac{x-1}{x^2+2}\ dx \ ,\ \int \frac{2x+1}{x^2+x-3}\ dx \ , \ \int \frac{x}{x^4+16}\ dx$$ $$ \ \int{2x^2-15x+33\over {(x+1)(x-5)}}\ dx \ , \int \frac{1}{x^2+x+1} \ dx \ ,\ \int {1\over {(x^2+1)^3}} \ dx .$$ %\begin{correction} %$$\int{5x-12\over {x(x-4)}}\ dx=2\ln|x-4|+3\ln|x|+c \ , $$ %$$ \int{37-11x\over{(x+1)(x-2)(x-3)}}\ dx= 4\ln|x+1|-5\ln|x-2|+\ln|x-3|+c \ ,$$ %$$ \int{6x-11\over {(x-1)^2}}\ dx=\frac{5}{x-1}+6\ln|x-1|+c \ ,$$ %$$\int \frac{dx}{x^2+2} = \frac{1}{\sqrt{2}} \arctan \frac{x}{\sqrt{2}}+c\ ,$$ %$$\int \frac{2x+1}{x^2+x-3}\,dx = \ln|x^2+x-3|+c \ ,$$ %$$\int \frac{x}{x^4+16} \ dx= \frac{1}{2} \int \frac{du}{u^2+16} = \frac18 \arctan(u/4)+c= \frac18 \arctan(x^2/4)+c\ ,$$ %$$ \int{2x^2-15x+33\over {(x+1)(x-5)}}\ dx = 2x-10\ln|x+1|+\frac43\ln|x-5|+c\ ,$$ %$$ \int \frac{x-1}{x^2+x+1} \ dx=\frac12\ln(x^2+x+1)-\sqrt{3}\arctan\left(\frac{2}{\sqrt{3}}(x+1/2)\right)+c \ ,$$ %$$\ \int {1\over {(x^2+1)^3}} \ dx =\frac38\arctan(x)+\frac38\frac{x}{x^2+1}+\frac14\frac{x}{(x^2+1)^2}+c.$$ %\end{correction} \end{exo} \section{Changement de variable} \begin{exo} Calculer les primitives suivantes : $$\int {dx\over {x^2+16}},\ \int {e^x\over {1+e^{2x}}}\ dx,\ \int {x\over \sqrt{1-x^4}}\ dx,\ \int {dx\over \sqrt{x-1}},\ \int {dx\over (1+x)\sqrt{x}}$$ $$\int {e^x\over \sqrt{16-e^{2x}}}\ dx,\ \int {\cos x\over \sqrt{9-\sin^2 x}}\ dx,\ \int {e^x\over \sqrt{4-e^{x}}}\ dx,\ \int {dx\over 49-4x^2},\ \int {dx\over x\sqrt{9-x^4}}$$ %$$\int {dx\over \sqrt{5-e^{2 x}}},\ \int {dx\over x\sqrt{x^2-1}},\ \int{2x-3\over \sqrt{4x^2-8x+3}}\ dx,\ \int(3x+1)\sqrt{-x^2-2x+3}\ dx.$$ \end{exo} \section{Int\'egration par parties} \begin{exo} Calculer les primitives suivantes : $$\int x\ln{x}\ dx ,\ \int x^2\ e^{-x}\ dx,\ \int \Arctan x\ dx, \ \int e^{3x}\cos 2x\ dx,$$ $$\ \int \sin x \ln\cos x\ dx , \ \int x^3 e^{-x^2}\ dx,\ \int x^3 \sh x\ dx,\ \int x^3 \cos x^2\ dx .$$ \end{exo} \section{Fonctions en sinus et cosinus} \begin{exo} Calculer les primitives suivantes : $$ \int (\cos x \cos 2x + \sin x \sin 3x)dx \quad ; \quad \int \cos x \sin^4x dx \quad ; \quad \int \cos^6x dx \quad ; $$ $$ \int \frac{\cos^3 x}{\sin^5x}dx \quad ; \quad \int \frac{\sin^3x}{1+\cos x}dx \quad ; \quad \int \frac{dx}{\cos^4x+\sin^4x} \quad ; \quad \int \frac{\cos x}{1+\sin 2x}dx \quad ; $$ $$ \ \int {\cos^3 x \over {\sqrt{1+\sin x }}}\ dx,\ \int {\sin 2x \over {\sqrt{1+\sin x }}}\ dx, \int {dx \over {2+\sin x }}, \ \int {dx \over {4\sin x-3\cos x}}.$$ \end{exo} \section{Calculs d'aires} \begin{exo} Calculer l'aire de chacun des domaines d\'efinis par les conditions suivantes : $$y\leq x^2+1,\quad y-x\geq 2,\ -2\leq x\le 2,\leqno(a)$$ $$0\leq y\leq \sqrt{x},\quad x+y\leq 6,\quad x \geq 1,\leqno(b)$$ $$y\leq x+3,\quad x\leq -y^2+3,\leqno(c)$$ $$0\leq x\leq 5\pi,\quad \ 3\sin {x\over2}\leq y\leq 4+\cos (2x) \,.\leqno(d)$$ \end{exo} %\begin{exo} %Calculer le volume de chacun des corps engendr\'es par rotation autour de %l'axe $Ox$ du domaine plan d\'efini par les conditions %d'\'equations : %$$-1\leq x\leq 1,\quad 0\leq y\leq x^2+1\leqno(a)$$ %$$1\leq x\leq 3,\quad\ 0\leq y\leq {1\over x}\leqno(b)$$ %$$0\leq x\leq \pi,\quad 0\leq y\leq 1+\cos x\leqno(c)$$ %\end{exo} %\begin{exo} % Calculer le volume d'une pyramide droite de hauteur $h$ et de base %un carr\'e de cot\'e $a$. %\end{exo} \section{Int\'egrales classiques} \begin{exo} Soit $I_{n} = \int_{0}^{1}{ (1-t^2)^n dt}$. \begin{enumerate} \item \'Etablir une relation de r\'ecurrence entre $I_{n}$ et $I_{n + 1}$. \item Calculer $I_{n}$. %\item En d\'eduire $\sum\limits_{k = 0}^{n}{\frac{ (-1)^k}{2k + 1}C_{n}^{k}}$. \end{enumerate} \end{exo} \begin{exo}[Int\'egrales de Wallis] Soit $I_{n} = \int_0^{\frac{\pi}2} \sin ^n tdt$. \begin{enumerate} \item \'Etablir une relation de r\'ecurrence entre $I_{n}$ et $I_{n + 2}$. \item En d\'eduire $I_{2p}$ et $I_{2p + 1}$. %\item Montrer que $ (I_{n})_{n \in \N}$ est d\'ecroissante et strictement positive. %\item En d\'eduire que $I_{n} \sim I_{n + 1}$. %\item Calculer $nI_{n}I_{n + 1}$. %\item Donner alors un \'equivalent simple de $I_{n}$. \end{enumerate} \end{exo} \pagebreak \noindent{\large\bf Universit\'e Henri Poincar\'e\hfill Facult\'e des Sciences et Technologies} \noindent{\sl {Licences Math\'ematiques, SPI et Informatique \hfill Automne 2010}} \vspace{0.5cm} \begin{center} {\Large Liste d'exercices n$^{\circ}$5 bis: Calcul de primitives } \end{center} \vskip0,5cm \vskip1cm \begin{exo} Calculer les primitives des fractions rationnelles suivantes. $$1.\quad {1 \over a^2+x^2}\ ,\quad 2.\quad {1 \over{(1+x^2)}^2}\ ,\quad 3. \quad {x^3 \over x^2-4}\ ,\quad 4. \quad {4x \over{(x-2)}^2}\quad 5. \quad {1 \over x^2+x+1}.$$ $$6.\quad {1 \over{(t^2+2t-1)}^2}\ ,\quad 7.\quad {3t+1 \over{(t^2-2t+10)}^2} \ ,\quad 8. \quad {3t+1 \over{t^2-2t+10}}\ ,\quad 9. \quad { 1 \over{t^3+1}}\ ,\quad 10. \quad {x^3+2 \over{(x+1)}^2}.$$ $$11.\quad{x+1 \over{x{(x-2)}^2}} \ ,\quad 12.\quad {(x^2-1)(x^3+3) \over2x+2x^2}\ ,\quad 13. \quad {x^2 \over{{(x^2+3)}^3 (x+1)}}\ ,\quad 14. \quad {x^7+x^3-4x-1 \over x{(x^2+1)}^2}.$$ $$\quad 15. \quad{3x^4-9x^3+12x^2-11x+7\over(x-1)^3(x^2+1)} \ , \quad16.\quad \frac{1}{1-x^2} \ ,\quad 17.\quad \frac{1}{x^3-7x+6}\ ,$$ $$\quad 18. \quad \frac{2x^4+3x^3+5x^2+17x+30}{x^3+8}\ ,\quad 19.\quad \frac{4x^2}{x^4-1}.$$ \begin{correction} R\'esultats valables sur chaque intervalle du domaine de d\'efinition. \begin{enumerate} \item ${1 \over x^2+a^2}$ est un \'el\'ement simple. Primitives~: ${1 \over a} \arctan({x \over a})+k$. \item ${1 \over{(1+x^2)}^2}$ est un \'el\'ement simple. Primitives~: ${1 \over2} \arctan x + {x \over2(1+x^2)}+k$. \item ${x^3 \over x^2-4}=x+{2 \over x-2}+{2 \over x+2}$. Primitives~: ${x^2 \over2}+ \ln(x^2-4)^2 +k$. \item ${4x \over{(x-2)}^2}={4 \over x-2}+{8 \over{(x-2)}^2}$. Primitives~: $4 \ln\vert x-2\vert-{8 \over x-2}+k$. \item ${1 \over x^2+x+1}$ est un \'el\'ement simple. Primitives~: ${2 \over\sqrt3} \arctan{(2x+1) \over\sqrt3}+k$. \item $ {1 \over{(t^2+2t-1)}^2} = {1 \over8{(t+1+ \sqrt2)}^2 }+ { \sqrt2 \over16(t+1+\sqrt2)} + {1 \over8{(t+1- \sqrt2)}^2}+ { - \sqrt2 \over16(t+1- \sqrt2)}$.\newline Primitives~: $-{t+1 \over4(t^2+2t-1)} +{\sqrt2 \over16} \ln \left\vert{t+1+ \sqrt2 \over t+1- \sqrt2 } \right\vert+ k$. \item $ {3t+1 \over{(t^2-2t+10)}^2}$ est un \'el\'ement simple.\newline Primitives~: $-{3 \over2(t^2-2t+10)} +{2(t-1) \over9(t^2-2t+10)} +{2 \over27} \arctan({t-1 \over3}) +k $. \item $ {3t+1 \over t^2-2t+10}$ est un \'el\'ement simple. Primitives~: ${3 \over2} \ln(t^2-2t+10) +{4 \over3} \arctan({t-1 \over3})+k$. \item ${1 \over t^3+1}={1 \over3(t+1)}-{t-2 \over3(t^2-t+1)}$. Primitives~: $ {1 \over3}\ln\vert t+1 \vert-{1 \over6} \ln( t^2-t+1) + {1 \over\sqrt3} \arctan({2t-1 \over\sqrt3}) +k$. \item $ {x^3+2 \over{(x+1)}^2}= x-2+{3 \over x+1}+{1 \over{(x+1)}^2}$. Primitives~: ${x^2 \over2}-2x+3 \ln\vert x+1 \vert-{1 \over x+1} +k$. \item ${x+1 \over x{(x-2)}^2}= {1 \over4x}- {1 \over4(x-2)} +{3 \over2{(x-2)}^2}$. Primitives~: $ {1 \over4} \ln\vert x \vert-{1 \over4} \ln\vert x-2 \vert -{3 \over2(x-2)} +k$. \item ${(x^2-1)(x^3+3) \over2x+2x^2}= {1 \over2} (x^3-x^2+3) -{3 \over2x} $. Primitives~: ${x^4 \over8}-{x^3 \over6}+{3x \over2}-{3 \over2}\ln\vert x \vert+k$. \item ${x^2 \over{(x^2+3)}^3(x+1)}={1 \over4^3(x+1)}+ {1-x \over4^3 (x^2+3)} + {1-x \over4^2 {(x^2+3)}^2} -{3(1-x) \over4{(x^2+3)}^3}$.\newline Primitives~: $-{x+3 \over4^2{(x^2+3)}^2}-{2x-3 \over3.2^5(x^2+3)} -{1 \over2^7}\ln(x^2+3) - {1 \over3 \sqrt3 \, 2^6} \arctan({x \over\sqrt3}) +{1 \over4^3}\ln\vert x+1 \vert+k$. \item ${x^7+x^3-4x-1 \over x{(x^2+1)}^2}= x^2-2-{1 \over x} +{x+4 \over x^2+1}+ {x-6 \over{(x^2+1)}^2}$.\newline Primitives~: ${x^3 \over3}-2x -\ln\vert x \vert+{1 \over2}\ln (1+x^2) +\arctan x -{6x+1 \over2(x^2+1)} +k$. \item ${3x^4-9x^3+12x^2-11x+7\over(x-1)^3(x^2+1)}={1\over (x-1)^3}-{2\over(x-1)^2}+{3\over x-1}-{1\over x^2+1}$.\newline Primitives~: $-{1/2\over(x-1)^2}+{2\over x-1}+3\ln\vert x-1\vert-\arctan x +k$. \item $ \int \frac{dx}{1-x^2} = \frac12\ln|x+1|-\frac12\ln|x-1|$. \item $\int \frac{1}{x^3-7x+6} \ dx=\int \frac{1}{20(x+3)} -\frac{1}{4(x-1)} +\frac{1}{5(x-2)} \ dx=\frac{1}{20}\ln\Bigl\vert\frac{(x-2)^4(x+3)}{(x-1)^5}\Bigr\vert+C$. \item $\int \frac{2x^4+3x^3+5x^2+17x+30}{x^3+8} \ dx = \int 2x+3+\frac{2}{x+2}+\frac{3x-1}{x^2-2x+4}\ dx =x^2+3x+\ln(x+2)^2 + \frac{3}{2}\ln(x^2-2x+4) + \frac{2}{\sqrt{3}}\arctan\frac{x-1}{\sqrt{3}}+C$. \item $\int \frac{4x^2}{x^4-1} \ dx= \int \frac{2}{x^2+1} - \frac{1}{x+1} +\frac{1}{x-1} \ dx=\ln\Bigl\vert \frac{x-1}{x+1}\Bigr\vert +2\arctan x +C$ \end{enumerate} \end{correction} \end{exo} %\begin{exo} %Calculer les primitives suivantes : %$$ %\int \frac{dx}{x+\sqrt{x-1}} \quad ; \quad %\int \frac{dx}{x\sqrt{x^2+x+1}} \quad ; \quad %\int \frac{x}{\sqrt{9+4x^4}}dx \quad ; %$$ %$$ %\int \frac{\sqrt[3]{x+1}-\sqrt{x+1}}{x+2}dx \quad ; \quad %\int \frac{x+1}{\sqrt{-4x^2+4x+1}}dx. %$$ %\end{exo} \begin{exo} Calculer les primitives des fonctions suivantes. $$1.\quad e^{\sin^2x} \sin2x \ ,\quad 2.\quad \cos^5x \ ,\quad (\ch{x})^3 \ ,\quad \cos^4x \quad (\sh{x})^4 .$$ $$3.\quad x^3 e^x \ ,\quad 4.\quad \ln x \ , \quad \arcsin{x} \ ,\quad 6. \quad \ch{x}\sin x \ , \quad 7. \quad {1 \over\sin x} .$$ $$8.\quad {e^{2x}\over\sqrt{e^x+1}} \ ,\quad 9.\quad e^{ax}\cos bx \ ,\quad 10. \quad \sqrt{x \over(1-x)^3}\ , \mbox{ pour } 0