921 lines
52 KiB
Plaintext
921 lines
52 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "9f2398e3",
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"metadata": {},
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"source": [
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"# Całka nieoznaczona"
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]
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},
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{
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"cell_type": "markdown",
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"id": "364327c1",
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"metadata": {},
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"source": [
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"## Cel zajęć\n",
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"\n",
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"Celem zajęć jest wprowadzenie pojęcia całki nieoznaczonej oraz nabycie praktycznej umiejętności obliczania całek za pomocą metody przez podstawienie i przez części."
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]
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},
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{
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"cell_type": "markdown",
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"id": "e0904246",
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"metadata": {},
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"source": [
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"## Definicja funkcji pierwotnej\n",
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"\n",
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">**Funkcją pierwotną** funkcji $f$ w przedziale $(a,b)$ nazywamy każdą taką funkcję $F$ określoną w $(a,b)$ taką, że $F'(x)=f(x)$ dla $x\\in(a,b)$."
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]
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},
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{
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"cell_type": "markdown",
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"id": "071003b9",
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"metadata": {
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"tags": []
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},
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"source": [
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"### Przykład funkcji pierwotnej\n",
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"\n",
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"Funkcje $F_1(x)=x^2$ i $F_2(x)=x^2+115$ są funkcjami pierwotnymi do funkcji $f(x)=2x$ w dowolnym przedziale $(a,b)$."
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]
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},
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{
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"cell_type": "markdown",
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"id": "0569050c",
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"metadata": {},
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"source": [
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"### Przykład trudniejszej funkcji pierwotnej \n",
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"\n",
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"Pokażemy, że funkcja $F(x)=\\ln\\left(x+\\sqrt{x^2+1}\\right)$ jest funkcją pierwotną funkcji $f(x)=\\frac{1}{\\sqrt{x^2+1}}$.\n",
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"Mamy\n",
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"\n",
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"$$\n",
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"\\begin{align*}\n",
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"F'(x)=\\left(\\ln\\left(x+\\sqrt{x^2+1}\\right)\\right)'=&\\frac{1}{x+\\sqrt{x^2+1}}\\cdot\\left(1+\\frac{x}{\\sqrt{x^2+1}}\\right)\\\\\n",
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"=&\\frac{1}{x+\\sqrt{x^2+1}}\\cdot \\frac{x+\\sqrt{x^2+1}}{\\sqrt{x^2+1}}\\\\=&\\frac{1}{\\sqrt{x^2+1}}.\n",
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"\\end{align*}\n",
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"$$"
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]
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},
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{
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"cell_type": "markdown",
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"id": "7bc55403",
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"metadata": {},
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"source": [
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"### Przykład funkcji nie posiadającej funkcji pierwotnej \n",
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"\n",
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"Pokażemy, że funkcja $f\\colon\\mathbb{R}\\to\\mathbb{R}$ dana wzorem $f(x)=1$, gdy $x \\geq 0$ i $f(x)=0$, gdy $x < 0$, nie posiada funkcji pierwotnej. \n",
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"\n",
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"Jeśli funkcja $F$ byłaby funkcją pierwtotną fukcji $f$, to dla $x>0$ musiałoby być $F(x)=x+C_1$ i $F(x)=C_2$ dla $x<0$, gdzie $C_1$ i $C_2$ są pewnymi stałymi. Ponieważ funkcja $F$ musi być ciągła w punkcie x=0, to musi być $C_1=C_2=F(0)$.\n",
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"Ale tak określona funkcja $F$ nie ma pochodnej w punkcie $x=0$, bo \n",
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"\n",
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"$$\n",
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"F'_-(0)=\\lim_{h\\to 0^-}\\frac{F(h)-F(0)}{h}=\\lim_{h\\to 0^-}\\frac{C_1-C_1}{h}=0\n",
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"$$\n",
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"\n",
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"i \n",
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"\n",
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"$$\n",
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"F'_+(0)=\\lim_{h\\to 0^+}\\frac{F(h)-F(0)}{h}=\\lim_{h\\to 0^+}\\frac{h+C_1-C_1}{h}=1.\n",
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"$$\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"id": "b141f464",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"image/png": 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\n",
|
|
"text/plain": [
|
|
"<Figure size 800x600 with 1 Axes>"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"import numpy as np\n",
|
|
"import matplotlib.pyplot as plt\n",
|
|
"\n",
|
|
"def custom_function(x):\n",
|
|
" if x < 0:\n",
|
|
" return 2\n",
|
|
" else:\n",
|
|
" return x + 2\n",
|
|
"\n",
|
|
"# Generate x values\n",
|
|
"x_values = np.linspace(-10, 10, 400) # Adjust the range and number of points as needed\n",
|
|
"\n",
|
|
"# Calculate corresponding y values\n",
|
|
"y_values = np.array([custom_function(x) for x in x_values])\n",
|
|
"\n",
|
|
"# Create the plot\n",
|
|
"plt.figure(figsize=(8, 6))\n",
|
|
"plt.plot(x_values, y_values, label=\"f(x)\", color='red')\n",
|
|
"plt.xlabel('x')\n",
|
|
"plt.ylabel('f(x)')\n",
|
|
"plt.title('Piecewise Function')\n",
|
|
"plt.legend()\n",
|
|
"plt.grid(True)\n",
|
|
"\n",
|
|
"# Add coordinate axes\n",
|
|
"plt.axhline(y=0, color='black', linewidth=2)\n",
|
|
"plt.axvline(x=0, color='black', linewidth=2)\n",
|
|
"\n",
|
|
"plt.show()\n"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "c0a9801d",
|
|
"metadata": {},
|
|
"source": [
|
|
"**Uwaga:** jest jasne, że dowolne dwie funkcje pierwotne do funkcji $f$ w przedziale $(a,b)$ mogą się różnić conajwyżej o stałą. "
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "25a8a4b4",
|
|
"metadata": {},
|
|
"source": [
|
|
"## Definicja całki nieoznaczonej\n",
|
|
"\n",
|
|
"> Rodzinę wszystkich funkcji pierwotnych funkcji $f$ nazywamy **całką nieoznaczoną** z $f$ i zapisujemy $\\int f(x)dx$.\n",
|
|
"Zatem jeśli $F$ jest dowolną funkcją pierwotną, to $\\int f(x)dx=F(x)+C$, gdzie $C$ jest dowolną stałą."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "7f3cfb0a",
|
|
"metadata": {},
|
|
"source": [
|
|
"### Wzory, które trzeba znać\n",
|
|
"\n",
|
|
">1. $ \\int x^adx=\\frac {\\displaystyle {x^{a+1}}}{\\displaystyle {a+1}}+C$,\n",
|
|
"$a\\neq -1$, $x>0$. Gdy $a$ jest liczbą naturalną, to zastrzeżenie $x>0$ odpada; \n",
|
|
"gdy $a$ jest liczbą całkowitą ujemną to zamiast $x>0$ wystarczy założyć $x\\neq 0$.\n",
|
|
">\n",
|
|
">\n",
|
|
">2. $\\ \\int \\frac {\\displaystyle {dx}}{\\displaystyle x}=\\ln \\mid x\\mid +C \\ , $ $x\\neq 0$ .\n",
|
|
">\n",
|
|
">\n",
|
|
">3. $\\ \\int e^xdx=e^x+C$ ,\n",
|
|
">\n",
|
|
">\n",
|
|
">4. $\\ \\int a^xdx=\\frac {\\displaystyle a^x}{\\displaystyle {\\ln a}}+C$, $a>0$, $a\\neq 1$.\n",
|
|
">\n",
|
|
">\n",
|
|
">5. $\\ \\int\\cos x dx=\\sin x +C$ ,\n",
|
|
">\n",
|
|
">\n",
|
|
">6. $\\ \\int \\sin xdx=-\\cos x+C$ ,\n",
|
|
">\n",
|
|
">\n",
|
|
">7. $\\ \\int \\frac {\\displaystyle {dx}}{\\displaystyle {\\cos^2x}}=\n",
|
|
"\\text {tg }x +C$, $\\cos x\\neq 0$ .\n",
|
|
">\n",
|
|
">\n",
|
|
">8. $\\ \\int \\frac {\\displaystyle {dx}}{\\displaystyle {\\sin ^2x}}= -c\\text {tg }x+C$, $\\sin x\\neq 0$.\n",
|
|
">\n",
|
|
">\n",
|
|
">9. $\\ \\int \\frac {\\displaystyle {dx}}{\\displaystyle {\\sqrt {1-x^2}}}= \\text{arc}\\sin x+C,-1<x<1$.\n",
|
|
">\n",
|
|
">\n",
|
|
">10. $\\ \\int \\frac {\\displaystyle {dx}}{\\displaystyle {1+x^2}}= \\text {arctg }x +C$.\n",
|
|
"\n"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "6efbdce3",
|
|
"metadata": {},
|
|
"source": [
|
|
"### Przykłady\n",
|
|
"Opierając się na wzorze 1 otrzymujemy:\n",
|
|
"\n",
|
|
"- $ \\int x^3dx=\\frac{x^4}{4}+C$;\n",
|
|
"\n",
|
|
"\n",
|
|
"- $ \\int x^5dx=\\frac{x^6}{6}+C$;\n",
|
|
"\n",
|
|
"\n",
|
|
"- $ \\int \\sqrt{x}dx=\\frac{2}{3}x^{\\frac{3}{2}}+C$;\n",
|
|
"\n",
|
|
"\n",
|
|
"- $ \\int \\frac{1}{x^2}dx=-\\frac{1}{x}+C$;\n",
|
|
"\n",
|
|
"\n",
|
|
"- $ \\int \\frac{1}{x^7}dx=-\\frac{1}{6x^6}+C$;\n",
|
|
"\n",
|
|
"\n",
|
|
"- $ \\int \\frac{1}{\\sqrt{x}}dx=2\\sqrt{x}+C$;\n",
|
|
"\n",
|
|
"\n",
|
|
"Opierając się na wzorze 4 otrzymujemy:\n",
|
|
"\n",
|
|
"- $ \\int 4^x dx=\\frac{4^x}{\\ln 4}+C$."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "3ee8f36a",
|
|
"metadata": {},
|
|
"source": [
|
|
"### Przykłady do samodzielnego rozwiązania\n",
|
|
"\n",
|
|
"Oblicz:\n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\int xdx,\\ \\int x^8dx, \\int \\sqrt[5]{x}dx, \\ \\int \\frac{1}{x^4}dx, \\ \\int \\frac{1}{x^8}dx, \\ \\int \\frac{1}{\\sqrt[3]{x}}dx.\n",
|
|
"$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "9b2bc358",
|
|
"metadata": {},
|
|
"source": [
|
|
"## Jak obliczać całki?"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "241db7e4",
|
|
"metadata": {},
|
|
"source": [
|
|
"### Arytmetyczne własności całki\n",
|
|
">__Twierdzenie.__ (o działaniach arytmetycznych na całkach nieoznaczonych)\n",
|
|
">\n",
|
|
">Jeżeli funkcje $f$ i $g$ są całkowalne w przedziale $I$ (otwartym lub domkniętym),\n",
|
|
"a $c$ jest dowolną liczbą, to funkcje $f+g$, $f-g$ i $cf$ są też całkowalne\n",
|
|
"w $I$ oraz\n",
|
|
">\n",
|
|
">$$ \\int [f(x)+g(x)] \\; dx= \\int f(x)dx+ \\int g(x) \\; dx ,$$\n",
|
|
">\n",
|
|
">$$ \\int [f(x)-g(x)] \\; dx= \\int f(x)dx- \\int g(x) \\; dx ,$$\n",
|
|
">\n",
|
|
">$$ \\int cf(x) \\; dx=c \\int f(x) \\; dx.$$\n",
|
|
"\n"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "66dab4da",
|
|
"metadata": {},
|
|
"source": [
|
|
"#### Przykłady\n",
|
|
"\n",
|
|
"1. \n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\int 1+x+x^2+x^3dx=x+\\frac{x^2}{2}+\\frac{x^3}{3}+\\frac{x^4}{4}+C.\n",
|
|
"$$\n",
|
|
"\n",
|
|
"2. \n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\int \\frac{4}{x}dx=4\\ln |x|+C.\n",
|
|
"$$\n",
|
|
"\n",
|
|
"3. \n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\int \\frac{(x+1)^2}{x^3}dx=\\int \\frac{1}{x}+\\frac{2}{x^2}+\\frac{1}{x^3}dx=\n",
|
|
"\\ln |x| -\\frac{2}{x}-\\frac{1}{2x^2}+C.\n",
|
|
"$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "08c3177c",
|
|
"metadata": {},
|
|
"source": [
|
|
"#### Przykłady do samodzielnego rozwiązania\n",
|
|
"\n",
|
|
"Oblicz:\n",
|
|
"\n",
|
|
"1. $\\displaystyle\n",
|
|
"\\int 2+\\sin x +\\cos xdx.\n",
|
|
"$\n",
|
|
"\n",
|
|
"\n",
|
|
"2. $\\displaystyle\n",
|
|
"\\int \\frac{(x-1)^3}{x}dx.\n",
|
|
"$\n",
|
|
"\n",
|
|
"3. $\\displaystyle\n",
|
|
"\\int 1+4x-5x^4 dx.\n",
|
|
"$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "cdc459d7",
|
|
"metadata": {},
|
|
"source": [
|
|
"### Całkowanie przez podstawienie\n",
|
|
"\n",
|
|
"> Jeżeli funkcje $f$ i $g$ są różniczkowalne (w odpowiednich przedziałach), to wzór na __pochodną funkcji złożonej__ mówi, że \n",
|
|
">\n",
|
|
">$$\n",
|
|
"(f\\circ g)'(x)=f'(g(x))g'(x).\n",
|
|
"$$\n",
|
|
">Zatem \n",
|
|
">\n",
|
|
">$$\n",
|
|
"\\int f'(g(x))g'(x)dx=f(g(x))+C.\n",
|
|
"$$\n",
|
|
"> Wzór ten zwany jest wzorem na ___całkowanie przez podstawienie___.\n",
|
|
"\n",
|
|
"\n",
|
|
"__Uwaga:__\n",
|
|
"\n",
|
|
"\n",
|
|
"Całkując przez podstawienie można postąpić w następujący sposób:\n",
|
|
"\n",
|
|
">1. W wyrażeniu $\\int f'(g(x))g'(x)dx$ zastępujemy $g(x)$ przez $t$ a $g'(x)dx$ przez $dt$ i otrzymujemy $\\int f'(t)dt$.\n",
|
|
">2. Ponieważ \n",
|
|
">\n",
|
|
">$$\n",
|
|
"\\int f'(t)dt=f(t)+C, \n",
|
|
"$$\n",
|
|
">\n",
|
|
">to zastępując teraz $t$ przez $g(x)$\n",
|
|
"otrzymujemy ostatecznie, że \n",
|
|
">\n",
|
|
">$$\n",
|
|
"\\int f'(g(x))g'(x)dx=f(g(x))+C.\n",
|
|
"$$\n"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "1a69a960",
|
|
"metadata": {},
|
|
"source": [
|
|
"#### Przykłady"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "994c5a2d",
|
|
"metadata": {},
|
|
"source": [
|
|
"1. Obliczymy\n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\int 3x^2e^{x^3}dx.\n",
|
|
"$$\n",
|
|
"\n",
|
|
"W tym celu podstawiamy $t=x^3$. Wtedy $dt=3x^2dx$.\n",
|
|
"Stąd\n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\int 3x^2e^{x^3}dx=\\int e^t dt=e^t+C=e^{x^3}+C.\n",
|
|
"$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "ff4dce28",
|
|
"metadata": {},
|
|
"source": [
|
|
"2. Obliczymy\n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\int \\frac{1}{2\\sqrt{x}}\\cos\\left(\\sqrt{x}\\right)dx.\n",
|
|
"$$\n",
|
|
"\n",
|
|
"W tym celu podstawiamy $t=\\sqrt{x}$. Wtedy $dt=\\frac{1}{2\\sqrt{x}}dx$.\n",
|
|
"Stąd\n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\int \\frac{1}{2\\sqrt{x}}\\cos\\left(\\sqrt{x}\\right)dx=\\int \\cos(t)dt=\\sin (t)+C =\\sin\\left(\\sqrt{x}\\right)+C.\n",
|
|
"$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "a06233c5",
|
|
"metadata": {},
|
|
"source": [
|
|
"3. Obliczymy\n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\int \\frac{\\ln^5x}{x}dx.\n",
|
|
"$$\n",
|
|
"\n",
|
|
"W tym celu podstawiamy $t=\\ln x$. Wtedy $dt=\\frac{1}{x}dx$.\n",
|
|
"Stąd\n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\int \\frac{\\ln^5x}{x}dx=\\int t^5dt=\\frac{t^6}{6}+C=\\frac{\\ln^6x}{6}+C.\n",
|
|
"$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "e9ce4da9",
|
|
"metadata": {},
|
|
"source": [
|
|
"4. Obliczymy\n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\int x\\sqrt{1+x^2}dx.\n",
|
|
"$$\n",
|
|
"\n",
|
|
"W tym celu podstawiamy $t=1+x^2$. Wtedy $dt=2xdx$.\n",
|
|
"Stąd\n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\int x\\sqrt{1+x^2}dx=\\frac{1}{2}\\int 2x\\sqrt{1+x^2}dx=\\frac{1}{2}\\int\\sqrt{t}dt=\\frac{1}{2}\\cdot\\frac{2}{3}t^{\\frac{3}{2}}+C=\n",
|
|
"\\frac{1}{3}\\sqrt{(1+x^2)^3}+C.\n",
|
|
"$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "dde18667",
|
|
"metadata": {},
|
|
"source": [
|
|
"5. Obliczymy\n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\int \\frac{x^3}{1+x^8}dx.\n",
|
|
"$$\n",
|
|
"\n",
|
|
"W tym celu podstawiamy $t=x^4$. Wtedy $dt=4x^3dx$.\n",
|
|
"Stąd\n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\int \\frac{x^3}{1+x^8}dx=\\frac{1}{4}\\int \\frac{1}{1+t^2}dt=\\frac{1}{4}\\operatorname{arctg}(t)+C=\\frac{1}{4}\\operatorname{arctg}(x^4)+C.\n",
|
|
"$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "8d8ee83e",
|
|
"metadata": {},
|
|
"source": [
|
|
"#### Przykłady do samodzielnego rozwiązania\n",
|
|
"\n",
|
|
"1. $\\displaystyle \\int \\cos xe^{\\sin x}dx$\n",
|
|
"\n",
|
|
"\n",
|
|
"2. $\\displaystyle \\int \\frac{1}{x}\\frac{1}{1+(\\ln x)^2}dx$\n",
|
|
"\n",
|
|
"\n",
|
|
"3. $\\displaystyle \\int\\frac{\\sqrt[3]{\\operatorname{tg} x}}{\\cos^2x} dx$\n",
|
|
"\n",
|
|
"\n",
|
|
"4. $\\displaystyle \\int \\frac{x}{\\sqrt{1+x^2}}dx$\n",
|
|
"\n",
|
|
"\n",
|
|
"5. $\\displaystyle \\int \\frac{e^{\\frac{1}{x}}}{x^2} dx$\n",
|
|
"\n",
|
|
"\n",
|
|
"6. $\\displaystyle \\int \\sin^4x\\cos x dx$\n",
|
|
"\n",
|
|
"\n",
|
|
"7. $\\displaystyle \\int\\frac{\\operatorname{arctg}(x)}{1+x^2} dx$\n",
|
|
"\n",
|
|
"\n"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "6c8b1218",
|
|
"metadata": {},
|
|
"source": [
|
|
"### Przydatna obserwacja"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "fae0f10d",
|
|
"metadata": {},
|
|
"source": [
|
|
">Ze wzoru na pochodną funkcji złożonej wynika, że jeśli \n",
|
|
">\n",
|
|
">$$\n",
|
|
"\\int f(x)dx=F(x)+C, \n",
|
|
"$$\n",
|
|
">\n",
|
|
">to \n",
|
|
">\n",
|
|
">$$\n",
|
|
"\\int f(ax+b)dx=\\frac{F(ax+b)}{a}+C.\n",
|
|
"$$\n"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "e78929c2",
|
|
"metadata": {},
|
|
"source": [
|
|
"#### Przykłady "
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "b22011fb",
|
|
"metadata": {},
|
|
"source": [
|
|
"1. $\\displaystyle\\int e^{4x+1}dx=\\frac{e^{4x+1}}{4}+C$.\n",
|
|
"\n",
|
|
"\n",
|
|
"2. $\\displaystyle\\int \\cos(2x+1)dx=\\frac{\\sin(2x+1)}{2}+C$.\n",
|
|
"\n",
|
|
"\n",
|
|
"3. $\\displaystyle\\int \\frac{1}{4x+6}dx=\\frac{\\ln|4x+6|}{4}+C$.\n",
|
|
"\n",
|
|
"\n",
|
|
"4. $\\displaystyle\\int \\frac{1}{(5x+6)^{11}}dx=-\\frac{1}{50(5x+6)^{10}}+C$.\n"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "01aba8c8",
|
|
"metadata": {},
|
|
"source": [
|
|
"#### Przykłady do samodzielnego rozwiązania"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "3c0b85c8",
|
|
"metadata": {},
|
|
"source": [
|
|
"1. $\\displaystyle\\int e^{-3x+1}dx$.\n",
|
|
"\n",
|
|
"\n",
|
|
"2. $\\displaystyle\\int \\sin(8x+1)dx$.\n",
|
|
"\n",
|
|
"\n",
|
|
"3. $\\displaystyle\\int \\frac{1}{7x+7}dx$.\n",
|
|
"\n",
|
|
"\n",
|
|
"4. $\\displaystyle\\int \\frac{1}{(8x+6)^{8}}dx$."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "6a023e55",
|
|
"metadata": {},
|
|
"source": [
|
|
"### I jeszcze jedna obserwacja"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "f9f0d9bf",
|
|
"metadata": {},
|
|
"source": [
|
|
">$$\n",
|
|
"\\int \\frac{f'(x)}{f(x)}dx=\\ln |f(x)|+C.\n",
|
|
"$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "3d5728e9",
|
|
"metadata": {},
|
|
"source": [
|
|
"#### Przykłady"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "6bd7a0c6",
|
|
"metadata": {},
|
|
"source": [
|
|
"1. $\\displaystyle \\int\\frac{x}{1+x^2}dx=\\frac{1}{2} \\int\\frac{2x}{1+x^2}dx=\\frac{\\ln(1+x^2)}{2}+C$.\n",
|
|
"\n",
|
|
"\n",
|
|
"2. $\\displaystyle \\int\\frac{\\cos x}{7+\\sin x}dx=\\ln(7+\\sin x)+C$."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "5f7c11b1",
|
|
"metadata": {},
|
|
"source": [
|
|
"### Całkowanie przez części\n",
|
|
"\n",
|
|
">**Twierdzenie.**\n",
|
|
">\n",
|
|
">Jeżeli funkcje $f$ i $g$ mają pochodne $f'$ i $g'$ ciągłe w przedziale $I$ \n",
|
|
"(otwartym lub domkniętym), to\n",
|
|
">\n",
|
|
">$$\n",
|
|
"\\int f(x)g'(x) \\; dx=f(x)g(x)-\\int f'(x)g(x) \\; dx \\ .\n",
|
|
"$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "f9646570",
|
|
"metadata": {},
|
|
"source": [
|
|
"#### Przykłady"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "912c50d2",
|
|
"metadata": {},
|
|
"source": [
|
|
"1. Obliczymy \n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\int xe^xdx.\n",
|
|
"$$\n",
|
|
"\n",
|
|
"Skorzystamy ze wzoru na całkowanie przez części biorąc: $f(x)=x$ i $g'(x)=e^x$. Wtedy \n",
|
|
"$f'(x)=1$ i $g(x)=e^x$ i mamy\n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\int xe^xdx=xe^x-\\int e^xdx=xe^x-e^x+C.\n",
|
|
"$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "e4c22485",
|
|
"metadata": {},
|
|
"source": [
|
|
"2. Obliczymy \n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\int (x^2+2x+1)e^{x+1}dx.\n",
|
|
"$$\n",
|
|
"\n",
|
|
"Skorzystamy ze wzoru na całkowanie przez części biorąc: $f(x)=x^2+2x+1$ i $g'(x)=e^{x+1}$. Wtedy \n",
|
|
"$f'(x)=2x+2$ i $g(x)=e^{x+1}$ i mamy\n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\int (x^2+2x+1)e^{x+1}dx =(x^2+2x+1)e^{x+1}-\\int (2x+2)e^{x+1}dx.\n",
|
|
"$$\n",
|
|
"\n",
|
|
"Pozostałą do obliczenia całkę $\\int (2x+2)e^{x+1}dx$ obliczymy ponownie przez części biorac $f(x)=2x+2$ i $g'(x)=e^{x+1}$.\n",
|
|
"Wtedy $f'(x)=2$ i $g(x)=e^{x+1}$ i mamy\n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\begin{align}\n",
|
|
"\\int (x^2+2x+1)e^{x+1}dx =&(x^2+2x+1)e^{x+1}-(2x+2)e^{x+1}+\\int 2e^{x+1}dx\\\\ =&\n",
|
|
"(x^2+2x+1)e^{x+1}-(2x+2)e^{x+1}+2e^{x+1}+C\n",
|
|
"\\\\ =&\n",
|
|
"(x^2+1)e^{x+1}+C.\n",
|
|
"\\end{align}\n",
|
|
"$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "ed002a9a",
|
|
"metadata": {},
|
|
"source": [
|
|
"__Uwaga:__\n",
|
|
"Tak można obliczyć każdą całkę postaci\n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\int p(x)e^{ax+b}dx,\n",
|
|
"$$\n",
|
|
"gdzie $p(x)$ jest wielomianem."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "28323ff3",
|
|
"metadata": {},
|
|
"source": [
|
|
"3. Obliczymy \n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\int x\\sin(x)dx.\n",
|
|
"$$\n",
|
|
"\n",
|
|
"Skorzystamy ze wzoru na całkowanie przez części biorąc: $f(x)=x$ i $g'(x)=\\sin (x)$. Wtedy \n",
|
|
"$f'(x)=1$ i $g(x)=-\\cos(x)$ i mamy\n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\int x\\sin(x)dx=-x\\cos(x)+\\int \\cos(x)dx=-x\\cos(x)+\\sin(x)+C.\n",
|
|
"$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "c82234a5",
|
|
"metadata": {},
|
|
"source": [
|
|
"4. Obliczymy \n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\int x^2\\cos(2x+1)dx.\n",
|
|
"$$\n",
|
|
"\n",
|
|
"Skorzystamy ze wzoru na całkowanie przez części biorąc: $f(x)=x^2$ i $g'(x)=\\cos(2x+1)$. Wtedy \n",
|
|
"$f'(x)=2x$ i $g(x)=\\frac{\\sin(2x+1)}{2}$ i mamy\n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\int x^2\\cos(2x+1)dx =\\frac{x^2}{2}\\sin(2x+1)-\\int x\\sin(2x+1)dx.\n",
|
|
"$$\n",
|
|
"\n",
|
|
"Pozostałą do obliczenia całkę $\\int x\\sin(2x+1)dx$ obliczymy ponownie przez części biorac $f(x)=x$ i $g'(x)=\\sin(2x+1)$.\n",
|
|
"Wtedy $f'(x)=1$ i $g(x)=-\\frac{\\cos(2x+1)}{2}$ i mamy\n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\begin{align}\n",
|
|
"\\int x^2\\cos(2x+1)dx =&\\frac{x^2}{2}\\sin(2x+1)+\\frac{x}{2}\\cos(2x+1)-\\int \\frac{\\cos(2x+1)}{2}dx \\\\ =&\n",
|
|
"\\frac{x^2}{2}\\sin(2x+1)+\\frac{x}{2}\\cos(2x+1)- \\frac{\\sin(2x+1)}{4}+C.\n",
|
|
"\\end{align}\n",
|
|
"$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "5424678d",
|
|
"metadata": {},
|
|
"source": [
|
|
"__Uwaga:__\n",
|
|
"Tak można obliczyć każdą całkę postaci\n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\int p(x)\\sin(ax+b)dx, \\int p(x)cos(ax+b)dx\n",
|
|
"$$\n",
|
|
"gdzie $p(x)$ jest wielomianem."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "9e997b3c",
|
|
"metadata": {},
|
|
"source": [
|
|
"5. Obliczymy \n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\int \\ln xdx.\n",
|
|
"$$\n",
|
|
"\n",
|
|
"Skorzystamy ze wzoru na całkowanie przez części biorąc: $f(x)=\\ln x$ i $g'(x)=1$. Wtedy \n",
|
|
"$f'(x)=\\frac{1}{x}$ i $g(x)=x$ i mamy\n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\int \\ln x dx=x\\ln x-\\int 1dx=x\\ln x -x+C.\n",
|
|
"$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "ae33fda7",
|
|
"metadata": {},
|
|
"source": [
|
|
"6. Obliczymy \n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\int x^8\\ln xdx.\n",
|
|
"$$\n",
|
|
"\n",
|
|
"Skorzystamy ze wzoru na całkowanie przez części biorąc: $f(x)=\\ln x$ i $g'(x)=x^8$. Wtedy \n",
|
|
"$f'(x)=\\frac{1}{x}$ i $g(x)=\\frac{x^9}{9}$ i mamy\n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\int \\ln x dx=\\frac{x^9\\ln x}{9}-\\int \\frac{x^8}{9}dx=\\frac{x^9\\ln x}{9}- \\frac{x^9}{81}+C.\n",
|
|
"$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "6a29acdc",
|
|
"metadata": {},
|
|
"source": [
|
|
"__Uwaga:__\n",
|
|
"Tak można obliczyć każdą całkę postaci\n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\int p(x)\\ln xdx,\n",
|
|
"$$\n",
|
|
"gdzie $p(x)$ jest wielomianem."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "83108cd9",
|
|
"metadata": {},
|
|
"source": [
|
|
"7. Obliczymy \n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\int \\cos(x)e^xdx.\n",
|
|
"$$\n",
|
|
"\n",
|
|
"\n",
|
|
"Skorzystamy ze wzoru na całkowanie przez części biorąc: $f(x)=\\cos(x)$ i $g'(x)=e^x$. Wtedy \n",
|
|
"$f'(x)=-\\sin(x)$ i $g(x)=e^x$ i mamy\n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\int \\cos(x)e^xdx =\\cos(x)e^x+\\int \\sin(x)e^xdx.\n",
|
|
"$$\n",
|
|
"\n",
|
|
"Pozostałą do obliczenia całkę $\\int \\sin(x)e^xdx$ obliczymy ponownie przez części biorac $f(x)=\\sin(x)$ i $g'(x)=e^x$.\n",
|
|
"Wtedy $f'(x)=\\cos (x)$ i $g(x)=e^x$ i mamy\n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\begin{align}\n",
|
|
"\\int \\cos(x)e^xdx =&\\sin(x)e^x+\\cos(x)e^x-\\int \\cos(x)e^x.\n",
|
|
"\\end{align}\n",
|
|
"$$\n",
|
|
"\n",
|
|
"Stąd \n",
|
|
"\n",
|
|
"$$\n",
|
|
"2\\int \\cos(x)e^xdx =\\sin(x)e^x+\\cos(x)e^x,\n",
|
|
"$$\n",
|
|
"\n",
|
|
"a zatem \n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\int \\cos(x)e^xdx =\\frac{\\sin(x)e^x+\\cos(x)e^x}{2}+C.\n",
|
|
"$$\n"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "261877c9",
|
|
"metadata": {},
|
|
"source": [
|
|
"#### Przykłady do samodzielnego rozwiązania"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "63578605",
|
|
"metadata": {},
|
|
"source": [
|
|
"1. $\\displaystyle \\int (x+3)e^{x+4}dx$.\n",
|
|
"\n",
|
|
"\n",
|
|
"2. $\\displaystyle\\int x^3e^{x}dx$.\n",
|
|
"\n",
|
|
"\n",
|
|
"3. $\\displaystyle\\int (x+2)\\cos(2x+1)dx$.\n",
|
|
"\n",
|
|
"\n",
|
|
"4. $\\displaystyle\\int x^2\\sin(x+3)dx$.\n",
|
|
"\n",
|
|
"\n",
|
|
"5. $\\displaystyle\\int x^6\\ln xdx$."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "a3250bf9",
|
|
"metadata": {},
|
|
"source": [
|
|
"__Uwaga.__\n",
|
|
"Oczywisćie nie każdą całkę można policzyć używając tylko metody całkowania przez cześci i przez podstawienie.\n",
|
|
"Ponadto istnieją też takie funkcje, których funkcji pierwotnej nie można przedstawić za pomocą funkcji elementarnych."
|
|
]
|
|
}
|
|
],
|
|
"metadata": {
|
|
"kernelspec": {
|
|
"display_name": "Python 3 (ipykernel)",
|
|
"language": "python",
|
|
"name": "python3"
|
|
},
|
|
"language_info": {
|
|
"codemirror_mode": {
|
|
"name": "ipython",
|
|
"version": 3
|
|
},
|
|
"file_extension": ".py",
|
|
"mimetype": "text/x-python",
|
|
"name": "python",
|
|
"nbconvert_exporter": "python",
|
|
"pygments_lexer": "ipython3",
|
|
"version": "3.11.3"
|
|
},
|
|
"toc": {
|
|
"base_numbering": 1,
|
|
"nav_menu": {},
|
|
"number_sections": false,
|
|
"sideBar": true,
|
|
"skip_h1_title": false,
|
|
"title_cell": "Table of Contents",
|
|
"title_sidebar": "Contents",
|
|
"toc_cell": false,
|
|
"toc_position": {},
|
|
"toc_section_display": true,
|
|
"toc_window_display": true
|
|
}
|
|
},
|
|
"nbformat": 4,
|
|
"nbformat_minor": 5
|
|
}
|