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Created a subdirectory to contain notebooks about Trigonometric Integ…
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…rals and added a notebook about techniques for integrating even and odd powers of Sine and Cosine.
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Tom Malone committed Feb 12, 2016
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163 changes: 163 additions & 0 deletions ...ues/Trigonometric_Integrals/.ipynb_checkpoints/Powers_of_Sine_and_Cosine-checkpoint.ipynb
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{
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"# Integrating Powers of Sine & Cosine\n",
"\n",
"Integrals involving powers of trigonometric functions cannot be computed using anti-derivative formulas or the Substitution Rule.\n",
"\n",
"What follows are strategies and specific techniques for integrating these types of functions.\n",
"\n",
"---\n",
"\n",
"Most of the techniques involve using __trigonometric identities__ to algebraically transform the integrand into something more easily integrated. Therefore, the following identities will be useful:\n",
"\n",
"<table style='width: 100%; margin-top: 0;'>\n",
"<tr>\n",
" <td style='width: 50%; vertical-align: top; border-top: none;'>\n",
" <h5>Pythagorean Identity</h5>\n",
" \n",
" <ul style='list-style-image: url(\"./arrow_bullet.png\");'>\n",
" <li>$\\cos^2(x) + \\sin^2(x) = 1 \\\\$</li>\n",
" <li>$\\sin^2(x) = 1 - \\cos^2(x) \\\\$</li>\n",
" <li>$\\cos^2(x) = 1 - \\sin^2(x) \\\\$</li>\n",
" <li>$\\sec^2(x) = 1 + \\tan^2(x) \\\\$</li>\n",
" <li>$\\csc^2(x) = 1 + \\cot^2(x)$</li>\n",
" </ul>\n",
" </td>\n",
" <td style='width: 50%; vertical-align: top; border-top: none;'>\n",
" <h5>Fundamental &amp; Reciprocal Identities</h5>\n",
" \n",
" <ul style='list-style-image: url(\"./arrow_bullet.png\");'>\n",
" <li>$\\csc(x) = \\frac{1}{\\sin(x)} \\\\$</li>\n",
" <li>$\\sec(x) = \\frac{1}{\\cos(x)} \\\\$</li>\n",
" <li>$\\tan(x) = \\frac{\\sin(x)}{\\cos(x)} = \\frac{1}{\\cot(x)} \\\\$</li>\n",
" <li>$\\cot(x) = \\frac{\\cos(x)}{\\sin(x)} = \\frac{1}{\\tan(x)}$</li>\n",
" </ul>\n",
" </td>\n",
"</tr>\n",
"<tr>\n",
" <td colspan='2' style='border-top: none;'>\n",
" <h5>Half-Angle Identities</h5>\n",
" \n",
" <table style='width: 100%; margin-top: 1rem;'>\n",
" <tr>\n",
" <td style='width: 50%; padding: 0; vertical-align: top; border-top: 0; margin: 0;'>\n",
" $\\cos^2(x) = \\frac{1}{2}\\left(1+\\cos(2x)\\right)$\n",
" </td>\n",
" <td style='width: 50%; padding: 0; vertical-align: top; border-top: 0; margin: 0;'>\n",
" $\\sin^2(x) = \\frac{1}{2}\\left(1-\\cos(2x)\\right)$\n",
" </td>\n",
" </tr>\n",
" </table>\n",
" </td>\n",
"</tr>\n",
"<tr>\n",
" <td colspan='2' style='border-top: none;'>\n",
" <h5>Double-Angle Identities</h5>\n",
" \n",
" <ul style='list-style-image: url(\"./arrow_bullet.png\");'>\n",
" <li>$\\sin(2x) = 2\\sin(x)\\cos(x) \\\\$</li>\n",
" <li>$\\cos(2x) = \\cos^2(x) - \\sin^2(x) \\quad = 2\\cos^2(x) - 1 \\quad = 1 - 2\\sin^2(x)$</li>\n",
" </ul>\n",
" </td>\n",
"</tr>\n",
"<tr>\n",
" <td colspan='2' style='border-top: none;'>\n",
" <h5>Negative-Angle Identities</h5>\n",
" \n",
" <table style='width: 100%; margin-top: 1rem;'>\n",
" <tr>\n",
" <td style='width: 50%; padding: 0; vertical-align: top; border-top: 0; margin: 0;'>\n",
" $\\sin(-x) = -\\sin(x)$\n",
" </td>\n",
" <td style='width: 50%; padding: 0; vertical-align: top; border-top: 0; margin: 0;'>\n",
" $\\cos(-x) = \\cos(x)$\n",
" </td>\n",
" </tr>\n",
" </table>\n",
" </td>\n",
"</tr>\n",
"</table>\n",
"\n",
"---\n",
"\n",
"## Integration Techniques\n",
"\n",
"### Odd Powers of Sine &amp; Cosine\n",
"\n",
"With odd powers of $\\sin$ and $\\cos$, you split off a single power of the function, rewriting, for instance, $\\cos^m(x)$ as $\\cos^{m-1}(x) \\cdot \\cos(x)$:\n",
"\n",
"$$\n",
"\\require{color}\n",
"\\int \\cos^5(x) dx \\quad = \\quad \\int \\cos^4(x) \\cdot \\cos(x) dx \\quad \\textcolor{gray} { Split \\space off \\space one \\space power} \\\\\n",
"= \\int \\left(1 - \\sin^2(x)\\right)^2\\cos(x) dx \\quad \\textcolor{gray} {Pythagorean \\space Identity} \\\\\n",
"= \\int (1 - u^2)^2 du \\quad \\textcolor{gray} {Let \\space u = \\sin(x). \\space du = \\cos(x)dx} \\\\\n",
"= \\int (1 - 2u^2 + u^4) du \\quad \\textcolor{gray} {Expand} \\\\\n",
"= u - \\frac{2}{3}u^3 + \\frac{1}{5}u^5 + C \\quad \\textcolor{gray} {Integrate} \\\\\n",
"= \\sin(x) - \\frac{2}{3}\\sin^3(x) + \\frac{1}{5}\\sin^5(x) + C \\quad \\textcolor{gray} {Replace \\space u \\space with \\space \\sin(x)}\n",
"$$\n",
"\n",
"### Even Powers of Sine &amp; Cosine\n",
"\n",
"With even powers of $\\sin$ and $\\cos$, use the __Half-Angle Identities__ to reduce the powers in the integrand:\n",
"\n",
"$$\n",
"\\int \\sin^4(x) dx = \\int \\left( \\sin^2(x) \\right)^2 dx \\quad \\textcolor{gray} {Rewrite \\space to \\space make \\space next \\space step \\space more \\space obvious} \\\\\n",
"= \\int \\left(\\frac{1}{2}(1 - \\cos(2x)) \\right)^2 dx \\quad \\textcolor{gray} {Half-Angle \\space Identity} \\\\\n",
"= \\int \\frac{1}{4}\\left(1 - 2\\cos(2x) + \\cos^2(2x)\\right) dx \\quad \\textcolor{gray} {Expand} \\\\\n",
"= \\frac{1}{4}\\int \\left(1 - 2\\cos(2x) + \\cos^2(2x)\\right) dx \\quad \\textcolor{gray} {Move \\space constant \\space out \\space of \\space integrand} \\\\\n",
"= \\frac{1}{4}\\int \\left(1 - 2\\cos(2x) + \\frac{1}{2}\\left(1+\\cos(2\\cdot2x\\right)\\right) dx \\quad \\textcolor{gray} {Use \\space \\textbf{Half Angle Formula} \\space again} \\\\\n",
"= \\frac{1}{4} \\int \\left(\\frac{3}{2} - 2\\cos(2x) + \\frac{1}{2}\\cos(4x)\\right) dx \\quad \\textcolor{gray} {Simplify} \\\\\n",
"= \\frac{3x}{8} - \\frac{1}{4}\\sin(2x) + \\frac{1}{32}\\sin(4x) + C \\quad \\textcolor{gray} {Evaluate \\space the \\space integrals} \n",
"$$\n",
"\n",
"---\n",
"\n",
"## Example 1\n",
"\n",
"Evaluate $\\int \\sin^3(x) dx$.\n",
"\n",
"$$\n",
"\\int \\sin^3(x) dx = \\int \\sin^2(x) \\cdot \\sin(x) dx \\quad \\textcolor{gray} {Split \\space off \\space one \\space power} \\\\\n",
"= \\int (1 - \\cos^2(x))\\sin(x)dx \\quad \\textcolor{gray}{ \\textbf{Half Angle Formula}} \\\\\n",
"= -\\int (1 - u^2) du \\quad \\textcolor{gray} {Let \\space u = \\cos(x). \\space -du = \\sin(x)dx} \\\\\n",
"= u - \\frac{1}{3}u^3 + C \\quad \\textcolor{gray} {Integrate} \\\\\n",
"\\int \\sin^3(x) dx \\space = \\space \\cos(x) - \\frac{\\cos^3(x)}{3} + C \\quad \\textcolor{gray} {Replace \\space u \\space with \\space \\cos(x)}\n",
"$$"
]
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163 changes: 163 additions & 0 deletions 07_Integration_Techniques/Trigonometric_Integrals/Powers_of_Sine_and_Cosine.ipynb
View file
Original file line number Diff line number Diff line change
@@ -0,0 +1,163 @@
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Integrating Powers of Sine &amp; Cosine\n",
"\n",
"Integrals involving powers of trigonometric functions cannot be computed using anti-derivative formulas or the Substitution Rule.\n",
"\n",
"What follows are strategies and specific techniques for integrating these types of functions.\n",
"\n",
"---\n",
"\n",
"Most of the techniques involve using __trigonometric identities__ to algebraically transform the integrand into something more easily integrated. Therefore, the following identities will be useful:\n",
"\n",
"<table style='width: 100%; margin-top: 0;'>\n",
"<tr>\n",
" <td style='width: 50%; vertical-align: top; border-top: none;'>\n",
" <h5>Pythagorean Identity</h5>\n",
" \n",
" <ul style='list-style-image: url(\"./arrow_bullet.png\");'>\n",
" <li>$\\cos^2(x) + \\sin^2(x) = 1 \\\\$</li>\n",
" <li>$\\sin^2(x) = 1 - \\cos^2(x) \\\\$</li>\n",
" <li>$\\cos^2(x) = 1 - \\sin^2(x) \\\\$</li>\n",
" <li>$\\sec^2(x) = 1 + \\tan^2(x) \\\\$</li>\n",
" <li>$\\csc^2(x) = 1 + \\cot^2(x)$</li>\n",
" </ul>\n",
" </td>\n",
" <td style='width: 50%; vertical-align: top; border-top: none;'>\n",
" <h5>Fundamental &amp; Reciprocal Identities</h5>\n",
" \n",
" <ul style='list-style-image: url(\"./arrow_bullet.png\");'>\n",
" <li>$\\csc(x) = \\frac{1}{\\sin(x)} \\\\$</li>\n",
" <li>$\\sec(x) = \\frac{1}{\\cos(x)} \\\\$</li>\n",
" <li>$\\tan(x) = \\frac{\\sin(x)}{\\cos(x)} = \\frac{1}{\\cot(x)} \\\\$</li>\n",
" <li>$\\cot(x) = \\frac{\\cos(x)}{\\sin(x)} = \\frac{1}{\\tan(x)}$</li>\n",
" </ul>\n",
" </td>\n",
"</tr>\n",
"<tr>\n",
" <td colspan='2' style='border-top: none;'>\n",
" <h5>Half-Angle Identities</h5>\n",
" \n",
" <table style='width: 100%; margin-top: 1rem;'>\n",
" <tr>\n",
" <td style='width: 50%; padding: 0; vertical-align: top; border-top: 0; margin: 0;'>\n",
" $\\cos^2(x) = \\frac{1}{2}\\left(1+\\cos(2x)\\right)$\n",
" </td>\n",
" <td style='width: 50%; padding: 0; vertical-align: top; border-top: 0; margin: 0;'>\n",
" $\\sin^2(x) = \\frac{1}{2}\\left(1-\\cos(2x)\\right)$\n",
" </td>\n",
" </tr>\n",
" </table>\n",
" </td>\n",
"</tr>\n",
"<tr>\n",
" <td colspan='2' style='border-top: none;'>\n",
" <h5>Double-Angle Identities</h5>\n",
" \n",
" <ul style='list-style-image: url(\"./arrow_bullet.png\");'>\n",
" <li>$\\sin(2x) = 2\\sin(x)\\cos(x) \\\\$</li>\n",
" <li>$\\cos(2x) = \\cos^2(x) - \\sin^2(x) \\quad = 2\\cos^2(x) - 1 \\quad = 1 - 2\\sin^2(x)$</li>\n",
" </ul>\n",
" </td>\n",
"</tr>\n",
"<tr>\n",
" <td colspan='2' style='border-top: none;'>\n",
" <h5>Negative-Angle Identities</h5>\n",
" \n",
" <table style='width: 100%; margin-top: 1rem;'>\n",
" <tr>\n",
" <td style='width: 50%; padding: 0; vertical-align: top; border-top: 0; margin: 0;'>\n",
" $\\sin(-x) = -\\sin(x)$\n",
" </td>\n",
" <td style='width: 50%; padding: 0; vertical-align: top; border-top: 0; margin: 0;'>\n",
" $\\cos(-x) = \\cos(x)$\n",
" </td>\n",
" </tr>\n",
" </table>\n",
" </td>\n",
"</tr>\n",
"</table>\n",
"\n",
"---\n",
"\n",
"## Integration Techniques\n",
"\n",
"### Odd Powers of Sine &amp; Cosine\n",
"\n",
"With odd powers of $\\sin$ and $\\cos$, you split off a single power of the function, rewriting, for instance, $\\cos^m(x)$ as $\\cos^{m-1}(x) \\cdot \\cos(x)$:\n",
"\n",
"$$\n",
"\\require{color}\n",
"\\int \\cos^5(x) dx \\quad = \\quad \\int \\cos^4(x) \\cdot \\cos(x) dx \\quad \\textcolor{gray} { Split \\space off \\space one \\space power} \\\\\n",
"= \\int \\left(1 - \\sin^2(x)\\right)^2\\cos(x) dx \\quad \\textcolor{gray} {Pythagorean \\space Identity} \\\\\n",
"= \\int (1 - u^2)^2 du \\quad \\textcolor{gray} {Let \\space u = \\sin(x). \\space du = \\cos(x)dx} \\\\\n",
"= \\int (1 - 2u^2 + u^4) du \\quad \\textcolor{gray} {Expand} \\\\\n",
"= u - \\frac{2}{3}u^3 + \\frac{1}{5}u^5 + C \\quad \\textcolor{gray} {Integrate} \\\\\n",
"= \\sin(x) - \\frac{2}{3}\\sin^3(x) + \\frac{1}{5}\\sin^5(x) + C \\quad \\textcolor{gray} {Replace \\space u \\space with \\space \\sin(x)}\n",
"$$\n",
"\n",
"### Even Powers of Sine &amp; Cosine\n",
"\n",
"With even powers of $\\sin$ and $\\cos$, use the __Half-Angle Identities__ to reduce the powers in the integrand:\n",
"\n",
"$$\n",
"\\int \\sin^4(x) dx = \\int \\left( \\sin^2(x) \\right)^2 dx \\quad \\textcolor{gray} {Rewrite \\space to \\space make \\space next \\space step \\space more \\space obvious} \\\\\n",
"= \\int \\left(\\frac{1}{2}(1 - \\cos(2x)) \\right)^2 dx \\quad \\textcolor{gray} {Half-Angle \\space Identity} \\\\\n",
"= \\int \\frac{1}{4}\\left(1 - 2\\cos(2x) + \\cos^2(2x)\\right) dx \\quad \\textcolor{gray} {Expand} \\\\\n",
"= \\frac{1}{4}\\int \\left(1 - 2\\cos(2x) + \\cos^2(2x)\\right) dx \\quad \\textcolor{gray} {Move \\space constant \\space out \\space of \\space integrand} \\\\\n",
"= \\frac{1}{4}\\int \\left(1 - 2\\cos(2x) + \\frac{1}{2}\\left(1+\\cos(2\\cdot2x\\right)\\right) dx \\quad \\textcolor{gray} {Use \\space \\textbf{Half Angle Formula} \\space again} \\\\\n",
"= \\frac{1}{4} \\int \\left(\\frac{3}{2} - 2\\cos(2x) + \\frac{1}{2}\\cos(4x)\\right) dx \\quad \\textcolor{gray} {Simplify} \\\\\n",
"= \\frac{3x}{8} - \\frac{1}{4}\\sin(2x) + \\frac{1}{32}\\sin(4x) + C \\quad \\textcolor{gray} {Evaluate \\space the \\space integrals} \n",
"$$\n",
"\n",
"---\n",
"\n",
"## Example 1\n",
"\n",
"Evaluate $\\int \\sin^3(x) dx$.\n",
"\n",
"$$\n",
"\\int \\sin^3(x) dx = \\int \\sin^2(x) \\cdot \\sin(x) dx \\quad \\textcolor{gray} {Split \\space off \\space one \\space power} \\\\\n",
"= \\int (1 - \\cos^2(x))\\sin(x)dx \\quad \\textcolor{gray}{ \\textbf{Half Angle Formula}} \\\\\n",
"= -\\int (1 - u^2) du \\quad \\textcolor{gray} {Let \\space u = \\cos(x). \\space -du = \\sin(x)dx} \\\\\n",
"= u - \\frac{1}{3}u^3 + C \\quad \\textcolor{gray} {Integrate} \\\\\n",
"\\int \\sin^3(x) dx \\space = \\space \\cos(x) - \\frac{\\cos^3(x)}{3} + C \\quad \\textcolor{gray} {Replace \\space u \\space with \\space \\cos(x)}\n",
"$$"
]
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