M:entetemax: titre="Calcul intégral (1)" taille="1.3"
SH:rm -f integration.mc

%S{Énoncé}

FICHIER:enonce.tex::
\everymath{\displaystyle}
Calculer les intégrales ou primitives suivantes:
\begin{multicols}{2}
 \begin{enumerate}
    \item  $\int_{1}^{27}\frac1{t\sqrt[3]{t}}dt$
    \item  $\int_{0}^2(1-\abs{x-1})^3 dx$
    \item  $\int_{0}^4\sqrt{2x+1}\ dx$
    \item  $\int_{0}^{1/2}x^3(1-x^2)^{5/2}dx$
    \item  $\int_{0}^1\frac1{\sqrt{9x^2+3}}\ dx$
    \item  $\int_{0}^{\frac{\pi}2}x^2\sin x e^x\ dx$
    \item  $\int\frac{\ln(\sin x)}{\cos^2x}\ dx$
    \item  $\int_{0}^{\frac{\pi}2}\frac1{3\tan x+2}\ dx$
    \item  $\int_{0}^{\pi}\frac1{a^2\cos^2x+\sin^2x}\ dx$
    \item  $\int\frac{\cos 2x}{\sin x+\sin 3x}\ dx$ 
	  {\footnotesize (CV: $t=\cos x$)}
    \item  $\int\frac{1-\cos 2x}{\sin 3x}\ dx$
          {\footnotesize (CV: $t=\cos x$)}
    \item  $\int\frac{dx}{2\ch x+\sh x+1}$
          {\footnotesize (CV: $t=\th\frac{x}2$)}
    \item  $\int\frac{\sqrt{x+1}-\sqrt[3]{x+1}}{\sqrt{x+1}+\sqrt[3]{x+1}}\ dx$\\
	  {\footnotesize (CV: $t=(x+1)^{1/6}$)}
    \item  $\int\frac{\sqrt{x^3+1}}x\ dx$ 
          {\footnotesize (CV: $\displaystyle t^2=x^3+1$)}
    \item  $\int\frac{x+1}{\sqrt{x(1-2x)}}\ dx$ 
          {\footnotesize (CV: $x-\frac14=\frac14\cos t$)}
    \item  $\int\sin2x\sh3x\ dx$
    \item  $\int \tan^5(x)\ dx$
    \item  $\int \frac{\sin x}{2+\tan^2x}\ dx$
    \item  $\int\frac1{\cos^4x+\sin^4x}\ dx$
 \end{enumerate}
\end{multicols}
§


M:texel: f="enonce" patron="latex"

%P{}

%S{Calculs}

FICHIER:liste.txt:
>load("integration.mc")$
>integre(1/(t*t^(1/3)),t,1,27);
>integre_chasles((1-abs(x-1))^3,x,0,1,2);
>integre(sqrt(2*x+1),x,0,4);
>integre(x^3*(1-x^2)^(5/2),x,0,1/2);
>integre(1/sqrt(9*x^2+3),x,0,1);
>integre(x^2*sin(x)*exp(x),x,0,%pi/2);
>primitive_simplifie(log(sin(x))/cos(x)^2,x,strig1,expand); 
>integre_limite(1/(3*tan(x)+2),x,0,%pi/2);
>block(assume(a>0),integre_chasles_limite(1/(a^2*cos(x)^2+sin(x)^2),x,0,%pi/2,%pi));
>primitive_simplifie(cos(2*x)/(sin(x)+sin(3*x)),x,strig2,radcan);
>primitive_simplifie((1-cos(2*x))/sin(3*x),x,strig2,radcan);
>primitive(1/(2*cosh(x)+sinh(x)+1),x);
>f(x):=((x+1)^(1/2)-(x+1)^(1/3))/((x+1)^(1/2)+(x+1)^(1/3));def
>block(assume(t>0),primitive_cv(f(x),x,t-(1+x)^(1/6),t));
>rhs(%)=ev(rhs(%),nouns);
>block(assume(t>0),primitive_cv(sqrt(x^3+1)/x,x,t^2-x^3-1,t));
>rhs(%)=ev(rhs(%),nouns);
>primitive((x+1)/sqrt(x*(1-2*x)),x);
>primitive(sin(2*x)*sinh(3*x),x);
>primitive_cv(tan(x)^5,x,cos(x)-t,t);
>rhs(%)=ev(rhs(%),nouns);
>primitive(sin(x)/(2+tan(x)^2),x);
>primitive(1/(cos(x)^4+sin(x)^4),x);
§

M:seq2fabA:  f="liste" 


%VTEX{\newpage\section{Macros: \texttt{integration.mc}}}
%ITEX{env=verbatim}{integration.mc}
%RM{*.tmp *.log  *.aux *.dvi *.tex}