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gucio321 · Dec 11, 2023 · dd6646e · dd6646e
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diff --git a/.doctrees/assets/notes/matematyka/matematyka_2023.12.11.doctree b/.doctrees/assets/notes/matematyka/matematyka_2023.12.11.doctree
diff --git a/.doctrees/environment.pickle b/.doctrees/environment.pickle
diff --git a/assets/notes/matematyka/matematyka_2023.12.11.html b/assets/notes/matematyka/matematyka_2023.12.11.html
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+
+
+<!DOCTYPE html>
+
+<html lang="pl" data-content_root="../../../">
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+    <title>Przez podstawienie &#8212; Matematyka  - dokumentacja</title>
+    <link rel="stylesheet" type="text/css" href="../../../_static/pygments.css?v=fa44fd50" />
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+  </head><body>
+    <div class="relbar-top">
+
+    <div class="related" role="navigation" aria-label="related navigation">
+      <h3>Nawigacja</h3>
+      <ul>
+        <li class="right" style="margin-right: 10px">
+          <a href="../../../genindex.html" title="Indeks ogólny"
+             accesskey="I">indeks</a></li>
+    <li><a href="../../../index.html">Matematyka  - dokumentacja</a> &#187;</li>
+
+        <li class="nav-item nav-item-this"><a href="">Przez podstawienie</a></li> 
+      </ul>
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+    </div>
+
+
+    <div class="document">
+      <div class="documentwrapper">
+        <div class="bodywrapper">
+          <div class="body" role="main">
+
+  <section id="przez-podstawienie">
+<h1>Przez podstawienie<a class="headerlink" href="#przez-podstawienie" title="Link to this heading">¶</a></h1>
+<div class="admonition-podstawienie admonition">
+<p class="admonition-title">Podstawienie</p>
+<p>Niech <span class="math notranslate nohighlight">\(G(y)\)</span> będzie funkcją pierwotną funkcji <span class="math notranslate nohighlight">\(g(y)\)</span> a <span class="math notranslate nohighlight">\(f(x)\)</span> będzie klasy <span class="math notranslate nohighlight">\(C^1\)</span>.
+Wtedy $\int g(y)dy |_{y = f(x)} = \int g(y) f’(x) dx</p>
+<div class="admonition tip">
+<p class="admonition-title">Wskazówka</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\int \sqrt{1-x^2} dx \\
+x = sin t \\
+dx = cos t dt \\
+\int \sqrt{1-sin^2t} cos t*dt = \\
+= \int cos^2(t) dt = \frac{1}{2} \int 2 cos^2{t} dt = \\
+= \frac{1}{2} \int 1 + cos(2t) dt = \\
+= \frac{1}{2} (t + \frac{1}{2} sin(2t)) = \\
+\frac{1}{2} arcsin(x) + \frac{1}{2}x \sqrt{1-x^2}\end{split}\]</div>
+</div>
+</div>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+I_n = \int \frac{1}{(1+x^2)^n} \\
+I_1 = arctg(x) \\
+I_n = \int \frac{1+x^2 - x^2}{1+x^2} dx = \int \frac{dx}{(1+x^2)^n} - \int \frac{x^2}{(1+x^2)^n}dx = \\
+= I_{n-1} - \int \frac{x^2}{(1+x^2)^n}
+I_n = \frac{1}{2n-2} \frac{x}{(1+x^2)^{n-1}} + \frac{2n-3}{2n-2} I_{n-1}
+\end{split}\]</div>
+</section>
+<section id="calki-funkcji-wymiernych">
+<h1>Całki funkcji wymiernych<a class="headerlink" href="#calki-funkcji-wymiernych" title="Link to this heading">¶</a></h1>
+<div class="math notranslate nohighlight">
+\[
+\int \frac{P(x)}{Q(x)}  = ?
+\]</div>
+<p>Należy znaleźć rozkład funkcji na ułamki proste:</p>
+<table class="docutils align-default">
+<thead>
+<tr class="row-odd"><th class="head"><p>f(x)</p></th>
+<th class="head"><p><span class="math notranslate nohighlight">\(\int f(x) dx\)</span></p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(\frac{A}{x-A}\)</span></p></td>
+<td><p>$ln(x-A)</p></td>
+</tr>
+<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(\frac{A}{(x-A)^k}\)</span></p></td>
+<td><p><span class="math notranslate nohighlight">\(\frac{-1}{x-1} ....\)</span></p></td>
+</tr>
+</tbody>
+</table>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(\frac{Cx - B}{(x-p)^2 + q^2}\)</span></p></li>
+<li><p><span class="math notranslate nohighlight">\(\frac{Cx - B}{((x-p)^2 + q^2)^k}\)</span></p></li>
+</ul>
+</section>
+
+
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+    <ul>
+<li><a class="reference internal" href="#">Przez podstawienie</a></li>
+<li><a class="reference internal" href="#calki-funkcji-wymiernych">Całki funkcji wymiernych</a></li>
+</ul>
+
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+  <h3 id="searchlabel">Szybkie wyszukiwanie</h3>
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+          <a href="../../../genindex.html" title="Indeks ogólny"
+             >indeks</a></li>
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diff --git a/objects.inv b/objects.inv
diff --git a/searchindex.js b/searchindex.js