From 7391229b73e3dc661443972a771b35f9efa8b8bb Mon Sep 17 00:00:00 2001 From: Tom Malone Date: Sun, 21 Feb 2016 17:26:32 -0500 Subject: [PATCH] Added difficult problems I had trouble solving from Section 7.4: Integration by Partial Fraction Decomposition. Solved the first problem and wrote the solution in LaTex. --- .../_DIFFICULT_PROBLEMS-checkpoint.ipynb | 79 ++++++++++++++++++- _DIFFICULT_PROBLEMS.ipynb | 79 ++++++++++++++++++- 2 files changed, 152 insertions(+), 6 deletions(-) diff --git a/.ipynb_checkpoints/_DIFFICULT_PROBLEMS-checkpoint.ipynb b/.ipynb_checkpoints/_DIFFICULT_PROBLEMS-checkpoint.ipynb index 6fc2914..c332b6b 100644 --- a/.ipynb_checkpoints/_DIFFICULT_PROBLEMS-checkpoint.ipynb +++ b/.ipynb_checkpoints/_DIFFICULT_PROBLEMS-checkpoint.ipynb @@ -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)
\n", + "The denominator is an _irreducible quadratic expression—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__
\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__
\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__
\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", + "
\n", + "---\n", + "
\n", + "\n", "2) $$\\int \\frac{x^3+36}{x^2+36} dx$$" ] }, { - "cell_type": "code", - "execution_count": null, + "cell_type": "markdown", "metadata": { "collapsed": true }, - "outputs": [], "source": [] } ], diff --git a/_DIFFICULT_PROBLEMS.ipynb b/_DIFFICULT_PROBLEMS.ipynb index 6fc2914..c332b6b 100644 --- a/_DIFFICULT_PROBLEMS.ipynb +++ b/_DIFFICULT_PROBLEMS.ipynb @@ -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)
\n", + "The denominator is an _irreducible quadratic expression—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__
\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__
\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__
\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", + "
\n", + "---\n", + "
\n", + "\n", "2) $$\\int \\frac{x^3+36}{x^2+36} dx$$" ] }, { - "cell_type": "code", - "execution_count": null, + "cell_type": "markdown", "metadata": { "collapsed": true }, - "outputs": [], "source": [] } ],