From a5743cd9d99beafee16354b755bc5aeeb03c2a67 Mon Sep 17 00:00:00 2001
From: "flyingscorpio@pinebookpro" <tfranken@protonmail.com>
Date: Tue, 4 Jan 2022 09:28:00 +0100
Subject: [PATCH] Start correction for an

---
 theorie-signal/exercices/tp1.tex | 22 +++++++++++++++-------
 1 file changed, 15 insertions(+), 7 deletions(-)

diff --git a/theorie-signal/exercices/tp1.tex b/theorie-signal/exercices/tp1.tex
index 3983b1c..bc43368 100644
--- a/theorie-signal/exercices/tp1.tex
+++ b/theorie-signal/exercices/tp1.tex
@@ -36,17 +36,25 @@ Théorie du signal --- TP1
 
     \begin{align*}
         a_n &= \frac{2}{T_0} \int_{(T_0)} \left(1-\frac{2}{T_0}|t|\right)\cos(n\omega_0 t) \dif t \\
-            &= \frac{4}{T_0} \int_0^{\frac{T_0}{2}} 1-\frac{2}{T_0}|t| \dif t \int_0^{\frac{T_0}{2}}\cos(n\omega_0 t) \dif t \\
-            &= \frac{4}{T_0}\left(\frac{T_0}{2} - \frac{2}{T_0}\left[\frac{t^2}{2}\right]_0^{\sfrac{T_0}{2}}\right) \left[\frac{\sin(n\omega_0 t)}{n\omega_0}\right]_0^{\sfrac{T_0}{2}} \\
-            &= \left(\frac{4T_0}{2T_0} - \frac{8T_0^2}{8T_0^2}\right) \frac{\sin(n\omega_0\frac{T_0}{2})}{n\omega_0} \\
-            &= 1 \cdot \frac{\sin(\frac{2\pi n}{2})}{\frac{2\pi n}{T_0}} \\
-            &= \frac{\sin(n\pi)}{\frac{2\pi n}{T_0}} \\
-        a_n &= 0
+        \text{IPP avec }
+            &\left\{
+        \begin{array}{l}
+            u = 1 - \frac{2}{T_0}|t| \quad\implies
+            u' = -\frac{2}{T_0} \\ \\
+            v' = \cos(n\omega_0 t) \quad\implies
+            v = \frac{\sin(n\omega_0 t)}{n\omega_0}
+            \\
+        \end{array}
+        \right.\\
+        a_n &= \frac{4}{T_0}\left[
+            1 - \frac{2}{T_0}|t|
+            \frac{\sin(n\omega_0 t)}{n\omega_0}
+        \right]_0^{T_0/2}
+        - \int_0^{\sfrac{T_0}{2}} -\frac{2}{T_0}\frac{\sin(n\omega_0 t)}{n\omega_0} \dif t
     \end{align*}
 
     \begin{align*}
         S_x(t) &= a_0 + \sum_{n=0}^{+\infty} a_n\cos(n\omega_0 t) \\
-               &= \frac{1}{2}
     \end{align*}
 
     \begin{align*}