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)<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) $$\\int \\frac{x^3+36}{x^2+36} dx$$"
    ]
   },
   {
-   "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)<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) $$\\int \\frac{x^3+36}{x^2+36} dx$$"
    ]
   },
   {
-   "cell_type": "code",
-   "execution_count": null,
+   "cell_type": "markdown",
    "metadata": {
     "collapsed": true
    },
-   "outputs": [],
    "source": []
   }
  ],