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\noindent 
Universit\'e Claude Bernard Lyon 1  
\hfill UE Fondamentaux des Math\'ematiques II\\
Semestre de printemps 2017-2018
\vspace{2cm}

\begin{center}
{\Large\bf Feuille 9 : Primitives} 
\end{center}
\vspace{2cm}

\noindent 
\underline{\bf Exercice 9-1} Primitives usuelles à connaître

$$\begin{array}{lll}
  a)\, \int (2x^3 - 3x^2 + 4) \, dx \quad &
  b)\, \int \sqrt{x} \, dx \quad &
  c)\, \int \frac{3}{2x+4} \, dx \quad \\
  d)\, \int \frac{2}{x^2+1} \, dx \quad &
  e)\, \int \frac{1}{\cos^2 x} \, dx \quad &
  f)\, \int \sin x \, dx \quad \\
  g)\, \int e^{3x+1} \, dx \quad &
  h)\, \int \frac{1}{\sqrt{x+1}} \, dx \quad &
  i)\, \int \frac{1}{\sqrt{1-x^2}} \, dx \quad \\
\end{array}$$

\vspace{0.5cm}
\noindent 
\underline{\bf Exercice 9-2} Primitives "décalées", à reconnaître

$$\begin{array}{lll}
  a)\, \int (x^2 - 3x + 4)^5 (4x-6) \, dx \quad &
  b)\, \int \sin^7 x \cos x \, dx \quad &
  c)\, \int \frac{\sin x}{\cos x}\, dx \quad \\
  d)\, \int \frac{9x^2 - 6}{x^3 - 2x +5}\, dx \quad &
  e)\, \int \frac{3x}{\sqrt{x^2 - 2}}\, dx \quad &
  f)\, \int e^{2x^2-4x} (x-1) \, dx \quad \\
  g)\, \int \frac{3}{1-\sin^2 x} \, dx \quad &
  h)\, \int \frac{2x}{\sqrt{2-x^2}} \, dx \quad &
  i)\, \int \frac{x}{(x^2-4)^4}\, dx \quad \\
\end{array}$$

\vspace{0.5cm}
\noindent 
\underline{\bf Exercice 9-3} Fractions rationnelles

$$\begin{array}{lll}
  a)\, \int \frac{-2}{2x-5} \, dx \quad &
  b)\, \int \frac{3}{x^2+4} \, dx \quad &
  c)\, \int \frac{2x-5}{2x^2+3} \, dx \quad \\
  d)\, \int \frac{4}{(x-1)^5} \, dx \quad &
  e)\, \int \frac{x+2}{x^2+2x+5} \, dx \quad &
  f)\, \int \frac{2}{x (x-1) (x-2)} \, dx \quad \\
  g)\, \int_3^7 \frac{x^4}{x^3-3x+2} \, dx \quad &
  h)\, \int \frac{x}{x^3-1} \, dx \quad &
  i)\, \int \frac{3x^2}{1-x^2} \, dx \quad \\
\end{array}$$


\vspace{0.5cm}
\noindent 
\underline{\bf Exercice 9-4} Fonctions circulaires

$$\begin{array}{lll}
  a)\, \int \frac{1}{\sin x \cos x} \, dx \quad &
  b)\, \int \frac{\cos^3 x}{\sin^4 x} \, dx \quad &
  c)\, \int \sin^3 x \cos^2 x \, dx \quad \\
  d)\, \int \sin^4x \, dx \quad &
  e)\, \int \cos (2x) \sin x \, dx \quad &
  f)\, \int \frac{1}{\cos^2 x \sin^2 x} \, dx \quad \\
  g)\, \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \frac{\cos^2 x}{\sin x} \, dx \quad &
  h)\, \int_{\frac{-\pi}{4}}^{\frac{-\pi}{6}} \frac{2}{\sin x} \, dx \quad &
  i)\, \int \frac{3\sin x}{1-\cos^2 x} \, dx \quad \\
\end{array}$$


\newpage
\noindent 
\underline{\bf Exercice 9-5} Intégration par parties

$$\begin{array}{lll}
  a)\, \int \ln x \, dx \quad &
  b)\, \int (3x+5) \, e^{2x} \, dx \quad &
  c)\, \int \sin (x) \, e^{-x} \, dx \quad \\
  d)\, \int (x^2-1) \ln (2x) \, dx \quad &
  e)\, \int \arctan x \, dx \quad &
  f)\, \int 3x^2 \ln (x^2+1) \, dx \quad \\
  g)\, \int  (x^2+1) \arctan x \, dx \quad &
  h)\, \int  \arcsin x \, dx \quad &
  i)\, \int  x \arctan x \, dx \quad \\
\end{array}$$

\vspace{0.5cm}
\noindent 
\underline{\bf Exercice 9-6} Changement de variables

$$\begin{array}{lll}
  a)\, \int \frac{dx}{\sqrt{1+e^x}} \quad &
  b)\, \int \frac{dx}{\sqrt{x}+\sqrt[3]{x}} \quad &
  c)\, \int \frac{dx}{1+\sqrt{x}} \quad \\
  d)\, \int \frac{dx}{\sqrt{1+x^2}} \quad &
  e)\, \int \frac{dx}{2 + \sin x} \quad &
  f)\, \int (2x+1)\sqrt{x+2} \, dx \quad \\
\end{array}$$

\vspace{0.5cm}
\noindent 
\underline{\bf Exercice 9-7} Calculez $\int \frac{dx}{x^2(x-1)}$.

A l'aide du changement de variable $x=\sqrt{\frac{t}{t+1}}$, déterminez les primitives de $$\frac{1}{2}\int \frac{dt}{t(t+1)(\sqrt{\frac{t}{t+1}}-\frac{t}{t+1})}$$

\vspace{0.5cm}
\noindent 
\underline{\bf Exercice 9-8} Pour tout entier $n$, on définit $(I_n)_{n\in \N}$ par $I_n=\int_{0}^{\frac{\pi}{2}} sin^n(x) \, dx$.
\begin{enumerate}
  \item Calculez $I_0$ et $I_1$.
  \item Montrez que $\forall n \in \N$,$\quad I_{n+2}=\frac{n+1}{n+2} \, I_{n}$
  \item Montrez que pour $n=2k$, on a : $\quad I_{2k}=\frac{1 \cdot 3 \cdot \ldots \cdot (2k-1)}{2 \cdot 4 \cdot \ldots  \cdot 2k} \cdot \frac{\pi}{2}$
  \item Montrez que pour $n=2k+1$, on a :  $\quad I_{2k+1}=\frac{2 \cdot 4 \cdot \ldots \cdot 2k}{3 \cdot 5 \cdot \ldots  \cdot (2k+1)}$
  \item Montrez que $(I_n)_{n\in \N}$ est une suite décroissante.
  \item Montrez que $\quad \lim_{k \to \infty} \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \ldots \cdot \frac{2k}{2k-1} \cdot \frac{2k}{2k+1}=\frac{\pi}{2}$ (Formule de Wallis).
\end{enumerate}

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