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Added difficult problems I had trouble solving from Section 7.4: Inte…
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…gration by Partial Fraction Decomposition. Solved the first problem and wrote the solution in LaTex.
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Tom Malone committed Feb 21, 2016
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79 changes: 76 additions & 3 deletions .ipynb_checkpoints/_DIFFICULT_PROBLEMS-checkpoint.ipynb
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"## Partial Fraction Decomposition\n",
"1) $$\\int \\frac{x+20}{x^2+26x+173} dx$$\n",
"\n",
"__Step 1): Complete the Square__ (Denominator)<br/>\n",
"The denominator is an _irreducible quadratic expression&mdash;it cannot be factored._\n",
"\n",
"Therefore, to make it more \"pliable,\" complete the square:\n",
"\n",
"$$Ax^2+Bx+C \\space = \\space Ax^2+Bx+\\left(\\frac{B}{2}\\right)^2+C-\\left(\\frac{B}{2}\\right)^2 \\\\ \\\\$$\n",
"\n",
"$$x^2+26x+173 \\space = \\space x^2+26x+\\left(\\frac{26}{2}\\right)^2+173-\\left(\\frac{26}{2}\\right)^2 \\\\ \\\\$$\n",
"\n",
"$$x^2+26x+173 \\space = \\space (x^2+26x+169)+173-169 \\\\ \\\\ $$\n",
"\n",
"$$x^2+26x+173 \\space = \\space (x+13)^2+4 \\\\ \\\\ $$\n",
"\n",
"__Step 2): Split the integrand into multiple integrals and integrate using appropriate techniques__<br/>\n",
"Completing the square will help in a later step, but it made sense to get it out of the way, first.\n",
"\n",
"Now, split the integrand by splitting up the rational function:\n",
"\n",
"$$\\int \\frac{x+20}{x^2+26x+173} dx \\space = \\space \\int \\frac{x+13}{x^2+26x+173} dx \\quad + \\quad \\int \\frac{7}{x^2+26x+173}dx$$\n",
"\n",
"__Step 3): Integrate, one integral at a time__<br/>\n",
"The first integral can be integrated using the __Substitution Rule.__\n",
"\n",
"$$\n",
"\\text{Integral 1:} \\quad \\int \\frac{x+13}{x^2+26x+173} dx \\space = \\space \\frac{1}{2} \\int \\frac{2(x+13)}{x^2+26x+173} dx \\\\ \\\\\n",
"= \\frac{1}{2} \\int \\frac{2x+26}{x^2+26x+173} dx \\\\ \\\\\n",
"= \\frac{1}{2} \\ln(x^2+26x+173) + C \\quad \\tiny\\text{Note: normally, the argument to the} \\space \\textbf{natural log function} \\space \\text{must be placed within an} \\space \\textbf{absolute value sign,} \\\\ \\tiny\\text{but} \\space x^2+26x+173 \\space \\text{cannot be negative, so the absolute value sign is unnecessary.}\n",
"$$\n",
"\n",
"$$ \\\\\n",
"\\text{Integral 2:} \\quad \\int \\frac{7}{x^2+26x+173} dx \\space = \\space 7\\int \\frac{1}{x^2+26x+173} dx \\\\ \\\\\n",
"= 7 \\int \\frac{1}{(x+13)^2+4} dx\n",
"$$\n",
"\n",
"$$\n",
"\\tiny\\text{The most appropriate integration technique for this integral will be} \\space \\textbf{Trigonometric Substitution.} \\\\\n",
"\\implies \\text{Let} \\space u = 2\\tan\\theta \\\\\n",
"\\implies \\quad du = 2\\sec^2\\theta d\\theta \\\\ \n",
"\\implies \\quad \\tan\\theta = \\frac{u}{2} \\\\\n",
"\\implies \\quad \\theta = \\arctan\\left(\\frac{u}{2}\\right)\n",
"$$\n",
"\n",
"$$\n",
"\\implies \\quad 7 \\int \\frac{1}{(x+13)^2+4} dx \\quad = \\quad 7 \\int \\frac{2\\sec^2\\theta}{(2\\tan\\theta)^2+4} d\\theta \n",
"\\\\\n",
"\\\\\n",
"\\implies 7 \\int \\frac{2\\sec^2\\theta}{4\\tan^2\\theta+4} d\\theta \\quad = \\quad 7 \\int \\frac{2\\sec^2\\theta}{4(\\tan^2\\theta+1} d\\theta\n",
"\\\\\n",
"\\\\\n",
"\\implies 7 \\int \\frac{2\\sec^2\\theta}{4\\sec^2\\theta} d\\theta\n",
"\\\\\n",
"\\\\\n",
"\\implies 7 \\int \\frac{1}{2} d\\theta\n",
"\\\\\n",
"\\\\\n",
"\\implies \\frac{7}{2}\\theta + C \\quad \\tiny\\text{Now, do the back substitution using the value for} \\space \\theta \\space \\tiny\\text{and for} \\space u \\space \\tiny\\text{found previously.}\n",
"\\\\\n",
"\\\\ \n",
"\\theta = \\arctan\\left(\\frac{u}{2}\\right) \\quad \\implies \\quad \\frac{7}{2}\\theta + C \\space = \\space \\frac{7}{2}\\arctan\\left(\\frac{x+13}{2}\\right) + C\n",
"$$\n",
"\n",
"__Step 4): Combine__<br/>\n",
"Combine the two antiderivatives to form the final solution.\n",
"$$\n",
"\\\\\n",
"\\int \\frac{x+20}{x^2+26x+173} dx = \\frac{1}{2}\\ln(x^2+26x+173) + \\frac{7}{2}\\arctan\\left(\\frac{x+13}{2}\\right) + C\n",
"\\\\\n",
"\\\\\n",
"\\small \\text{Done!}\n",
"$$\n",
"\n",
"<br/>\n",
"---\n",
"<br/>\n",
"\n",
"2) $$\\int \\frac{x^3+36}{x^2+36} dx$$"
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79 changes: 76 additions & 3 deletions _DIFFICULT_PROBLEMS.ipynb
View file
Original file line number Diff line number Diff line change
Expand Up @@ -10,16 +10,89 @@
"## Partial Fraction Decomposition\n",
"1) $$\\int \\frac{x+20}{x^2+26x+173} dx$$\n",
"\n",
"__Step 1): Complete the Square__ (Denominator)<br/>\n",
"The denominator is an _irreducible quadratic expression&mdash;it cannot be factored._\n",
"\n",
"Therefore, to make it more \"pliable,\" complete the square:\n",
"\n",
"$$Ax^2+Bx+C \\space = \\space Ax^2+Bx+\\left(\\frac{B}{2}\\right)^2+C-\\left(\\frac{B}{2}\\right)^2 \\\\ \\\\$$\n",
"\n",
"$$x^2+26x+173 \\space = \\space x^2+26x+\\left(\\frac{26}{2}\\right)^2+173-\\left(\\frac{26}{2}\\right)^2 \\\\ \\\\$$\n",
"\n",
"$$x^2+26x+173 \\space = \\space (x^2+26x+169)+173-169 \\\\ \\\\ $$\n",
"\n",
"$$x^2+26x+173 \\space = \\space (x+13)^2+4 \\\\ \\\\ $$\n",
"\n",
"__Step 2): Split the integrand into multiple integrals and integrate using appropriate techniques__<br/>\n",
"Completing the square will help in a later step, but it made sense to get it out of the way, first.\n",
"\n",
"Now, split the integrand by splitting up the rational function:\n",
"\n",
"$$\\int \\frac{x+20}{x^2+26x+173} dx \\space = \\space \\int \\frac{x+13}{x^2+26x+173} dx \\quad + \\quad \\int \\frac{7}{x^2+26x+173}dx$$\n",
"\n",
"__Step 3): Integrate, one integral at a time__<br/>\n",
"The first integral can be integrated using the __Substitution Rule.__\n",
"\n",
"$$\n",
"\\text{Integral 1:} \\quad \\int \\frac{x+13}{x^2+26x+173} dx \\space = \\space \\frac{1}{2} \\int \\frac{2(x+13)}{x^2+26x+173} dx \\\\ \\\\\n",
"= \\frac{1}{2} \\int \\frac{2x+26}{x^2+26x+173} dx \\\\ \\\\\n",
"= \\frac{1}{2} \\ln(x^2+26x+173) + C \\quad \\tiny\\text{Note: normally, the argument to the} \\space \\textbf{natural log function} \\space \\text{must be placed within an} \\space \\textbf{absolute value sign,} \\\\ \\tiny\\text{but} \\space x^2+26x+173 \\space \\text{cannot be negative, so the absolute value sign is unnecessary.}\n",
"$$\n",
"\n",
"$$ \\\\\n",
"\\text{Integral 2:} \\quad \\int \\frac{7}{x^2+26x+173} dx \\space = \\space 7\\int \\frac{1}{x^2+26x+173} dx \\\\ \\\\\n",
"= 7 \\int \\frac{1}{(x+13)^2+4} dx\n",
"$$\n",
"\n",
"$$\n",
"\\tiny\\text{The most appropriate integration technique for this integral will be} \\space \\textbf{Trigonometric Substitution.} \\\\\n",
"\\implies \\text{Let} \\space u = 2\\tan\\theta \\\\\n",
"\\implies \\quad du = 2\\sec^2\\theta d\\theta \\\\ \n",
"\\implies \\quad \\tan\\theta = \\frac{u}{2} \\\\\n",
"\\implies \\quad \\theta = \\arctan\\left(\\frac{u}{2}\\right)\n",
"$$\n",
"\n",
"$$\n",
"\\implies \\quad 7 \\int \\frac{1}{(x+13)^2+4} dx \\quad = \\quad 7 \\int \\frac{2\\sec^2\\theta}{(2\\tan\\theta)^2+4} d\\theta \n",
"\\\\\n",
"\\\\\n",
"\\implies 7 \\int \\frac{2\\sec^2\\theta}{4\\tan^2\\theta+4} d\\theta \\quad = \\quad 7 \\int \\frac{2\\sec^2\\theta}{4(\\tan^2\\theta+1} d\\theta\n",
"\\\\\n",
"\\\\\n",
"\\implies 7 \\int \\frac{2\\sec^2\\theta}{4\\sec^2\\theta} d\\theta\n",
"\\\\\n",
"\\\\\n",
"\\implies 7 \\int \\frac{1}{2} d\\theta\n",
"\\\\\n",
"\\\\\n",
"\\implies \\frac{7}{2}\\theta + C \\quad \\tiny\\text{Now, do the back substitution using the value for} \\space \\theta \\space \\tiny\\text{and for} \\space u \\space \\tiny\\text{found previously.}\n",
"\\\\\n",
"\\\\ \n",
"\\theta = \\arctan\\left(\\frac{u}{2}\\right) \\quad \\implies \\quad \\frac{7}{2}\\theta + C \\space = \\space \\frac{7}{2}\\arctan\\left(\\frac{x+13}{2}\\right) + C\n",
"$$\n",
"\n",
"__Step 4): Combine__<br/>\n",
"Combine the two antiderivatives to form the final solution.\n",
"$$\n",
"\\\\\n",
"\\int \\frac{x+20}{x^2+26x+173} dx = \\frac{1}{2}\\ln(x^2+26x+173) + \\frac{7}{2}\\arctan\\left(\\frac{x+13}{2}\\right) + C\n",
"\\\\\n",
"\\\\\n",
"\\small \\text{Done!}\n",
"$$\n",
"\n",
"<br/>\n",
"---\n",
"<br/>\n",
"\n",
"2) $$\\int \\frac{x^3+36}{x^2+36} dx$$"
]
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"cell_type": "code",
"execution_count": null,
"cell_type": "markdown",
"metadata": {
"collapsed": true
},
"outputs": [],
"source": []
}
],
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