{
  "cells": [
     {"cell_type":"markdown","source":"<h1>Questions to be handed in for project Techniques of Integration</h1>","metadata":{"internals":{"slide_type":"subslide","slide_helper":"subslide_end"},"slideshow":{"slide_type":"slide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<p>Begin by loading our package for plotting our package that allows symbolic math:</p>","metadata":{}},
{"outputs":[],"cell_type":"code","source":["using Plots\nusing SymPy"],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>This project covers some of the techniques of integration utilizing the symbolic algebra capabilities provide by the <code>SymPy</code> package. </p>","metadata":{}},
{"cell_type":"markdown","source":"<p>The basic techniques of integration involve </p>","metadata":{}},
{"cell_type":"markdown","source":"<ul>\n<li><p>substitution (reverse chain rule)</p>\n</li>\n<li><p>integration by parts (reversed product rule)</p>\n</li>\n<li><p>partial fractions (for rational functions)</p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"<p>The <code>SymPy</code> function <code>integrate</code> is a implementation of the <a href=\"http://en.wikipedia.org/wiki/Risch_algorithm\">Risch Algorithm</a>.  This algorithm can decide if a function comprised of elementary functions has an antiderivative expressed in elementary function.  For example, neither $\\sin(x^2)$ or $e^{x^2}$ has such a \"nice\" antiderivative, whereas $\\sin(x)^2$ and $xe^{-x^2}$ do. Risch's algorithm exploits <a href=\"http://asmeurersympy.wordpress.com/2010/08/\">Liouville's Theorem</a> which gives a representation for any potential antiderivative of $f(x)$ using <em>elementary functions</em> such as polynomials, logarithms, exponentials and trigonometric functions.</p>","metadata":{}},
{"cell_type":"markdown","source":"<h3>Questions</h3>","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<h4>When <code>integrate</code> works</h4>","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<ul>\n<li><p>use <code>integrate</code> to find an antiderivative for </p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"\n$$\n\\int \\cos(x) \\log(\\sin(x)) dx.\n$$\n","metadata":{}},
{"cell_type":"markdown","source":"<p>Judging from the answer does it appear to use parts, substitution, both, or something else.</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<ul>\n<li><p>use <code>integrate</code> to find an antiderivative for </p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"\n$$\n\\int (\\log(x))^2 dx.\n$$\n","metadata":{}},
{"cell_type":"markdown","source":"<p>Judging from the answer does it appear to use parts, substitution, both, or something else.</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<ul>\n<li><p>use <code>integrate</code> to find an antiderivative for </p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"\n$$\n\\int \\sin(x) \\cdot e^x dx.\n$$\n","metadata":{}},
{"cell_type":"markdown","source":"<p>Judging from the answer does it appear to use parts, substitution, both, or something else.</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<ul>\n<li><p>The <code>integrate</code> function might fail on</p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"\n$$\n\\int x \\sec^2(x) dx.\n$$\n","metadata":{}},
{"cell_type":"markdown","source":"<p>But, replacing $\\sec(x)$ with $1/\\cos(x)$ may work. Does it?</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<h3>It doesn't always work ...</h3>","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<p>The <code>integrate</code> function works for many functions, but for some one can help by performing substitutions by hand first.</p>","metadata":{}},
{"cell_type":"markdown","source":"<p>Here is an example from <a href=\"http://faculty.uml.edu/jpropp/142/Integration.pdf\">here</a> where performing the substitution by hand is needed to give a boost. Find</p>","metadata":{}},
{"cell_type":"markdown","source":"\n$$\n\\int (1 + \\log(x)) \\cdot \\sqrt{1 + (x\\cdot\\log(x))^2} dx.\n$$\n","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":["$$\\sqrt{x^{2} \\log{\\left (x \\right )}^{2} + 1} \\left(\\log{\\left (x \\right )} + 1\\right)$$"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["@vars x dx\nf(x) = sqrt(x^2 + 1)\nu = x*log(x)\nex = f(u) * diff(u,x)"],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>This is written to look like a substitution problem. Will <code>SymPy</code> be able to see that?</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":["$$\\int \\sqrt{x^{2} \\log{\\left (x \\right )}^{2} + 1} \\left(\\log{\\left (x \\right )} + 1\\right)\\, dx$$"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["integrate(ex, x)"],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>Sympy can't do this integral, and so returns the expression. We can help.</p>","metadata":{}},
{"cell_type":"markdown","source":"<p>Let's introduce a variable <code>dx</code> into our expression, and then we have:</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":["$$dx \\sqrt{x^{2} \\log{\\left (x \\right )}^{2} + 1} \\left(\\log{\\left (x \\right )} + 1\\right)$$"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["ex =  f(u) * diff(u,x) * dx"],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>Then we can have this substitution $u = x\\log(x)$  Then $du = (x \\cdot \\log(x))' dx$.</p>","metadata":{}},
{"cell_type":"markdown","source":"<p>We replace each $x \\log(x)$ with a <code>u</code>, and replace $dx$ with $dy/(x \\cdot \\log(x))'$. The <code>subs</code> function can do so:</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":["$$du \\sqrt{u^{2} + 1}$$"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["u, du = symbols(\"u, du\")\n## replace x*log(x) with u and diff(x*log(x),x) * dx with du\nex1 = subs(ex, (x*log(x), u), (dx, du/diff(x*log(x),x)))"],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>This latter function can be integrated (after stripping off the differential we added for familiarity):</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":["$$\\frac{u \\sqrt{u^{2} + 1}}{2} + \\frac{\\operatorname{asinh}{\\left (u \\right )}}{2}$$"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["ex = integrate(ex1 / du)"],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>If needed we can resubstitute in $x \\log(x)$ for $u$ to get an answer.</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":["$$\\frac{x \\sqrt{x^{2} \\log{\\left (x \\right )}^{2} + 1} \\log{\\left (x \\right )}}{2} + \\frac{\\operatorname{asinh}{\\left (x \\log{\\left (x \\right )} \\right )}}{2}$$"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["subs(ex, (u, x*log(x)))"],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>For future use, we codify the above steps in a function:</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/plain":["usub (generic function with 1 method)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["function usub(ex, let_u_equal)\n    u, du, dx = symbols(\"u, du, dx\")\n    ex1 = ex * dx\n    ex2 = subs(ex1, (dx, du/diff(let_u_equal, x)))\n    ex3 = subs(ex2, (let_u_equal, u))\n    ex3 / du\nend"],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>This can be used as follow:</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":["$$u$$"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["ex = log(x)/x\nusub(ex, log(x))"],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>While we are here, we also give this quick function for integration by parts. Just pick \"u\" and let <code>SymPy</code> do the ret:</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":["\\begin{bmatrix}\\frac{x^{2} \\sin{\\left (x \\right )}}{2}\\\\\\frac{x^{2} \\cos{\\left (x \\right )}}{2}\\end{bmatrix}"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["function udv_parts(ex, let_u_equal)\n    u, dv = let_u_equal, ex/let_u_equal\n    du = diff(u, x)\n    v = integrate(dv)\n    [u*v, v*du]\t\t\t# return two pieces. One for FTC, one to integrate\nend\n\nex = x*sin(x)\nuv, vdu = udv_parts(ex, sin(x))"],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<h3>Questions</h3>","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<h4>The absolute value</h4>","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<ul>\n<li><p>Does <code>integrate</code> know how to integrate $|x|$?</p>\n</li>\n</ul>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<ul>\n<li><p>Assert to <code>SymPy</code> that $x > 0$ by defining it via:</p>\n</li>\n</ul>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":["$$x$$"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["x = symbols(\"x\", positive=true)"],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>Does <code>integrate</code> now know how to integrate $|x|$?</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<ul>\n<li><p>What is a simple antiderivative for $\\int |x| dx$?</p>\n</li>\n</ul>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<h4>Helping <code>integrate</code> out</h4>","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<ul>\n<li><p>Does <code>integrate</code>  find an antiderivative for </p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"\n$$\n\\int \\log(\\log(x)) / x dx?\n$$\n","metadata":{}},
{"cell_type":"markdown","source":"<p>If not, help it out.</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<ul>\n<li><p>The following integral exists, but is not found by <code>integrate</code>:</p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"\n$$\n\\int \\frac{(-\\log(x)^2 + 1)^{1/2}}{x} dx\n$$\n","metadata":{}},
{"cell_type":"markdown","source":"<p>Make a $u$-substitution to help <code>integrate</code> out with finding an answer:</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<ul>\n<li><p>The following integral exists, but is not found by <code>integrate</code>:</p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"\n$$\n\\int \\frac{1}{x \\sqrt{1 - \\log(x)^2}}dx.\n$$\n","metadata":{}},
{"cell_type":"markdown","source":"<p>Make a $u$-substitution to help <code>integrate</code> out with finding an answer:</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<ul>\n<li><p>What do you get if you try a \"u\"-\"dv\" integration by parts for $\\int e^x \\sin(x) dx$?  Does it seem to help find the integral?</p>\n</li>\n</ul>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<ul>\n<li><p>The <a href=\"http://en.wikipedia.org/wiki/Integration_by_parts#LIATE_rule\">LIATE rule</a> is a rule of thumb for identifying what $u$ should be in integrating by parts. Apply it to </p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"\n$$\n\\int x^2 e^x dx.\n$$\n","metadata":{}},
{"cell_type":"markdown","source":"<p>What is $u$? What is the \"$v \\cdot du$\" term? Does it seem \"easier\" than what you started with? Why?</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<h3>Integration of rational functions:</h3>","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<p>Some background reading on the implementation is <a href=\"http://asmeurersympy.wordpress.com/2010/06/11/integration-of-rational-functions/\">here</a></p>","metadata":{}},
{"cell_type":"markdown","source":"<p>A <em>rational function</em> is a ratio of polynomial functions. Using polynomial long division and dividing out common factors, up to removable singularities, one can uniquely write a rational function in terms of three other polynomials:</p>","metadata":{}},
{"cell_type":"markdown","source":"\n$$\na(x)/b(x) =  s(x) + r(x)/q(x)\n$$\n","metadata":{}},
{"cell_type":"markdown","source":"<p>where the degree of $r(x)$ is less than the degree of $q(x)$ and there are no common roots to $r(x)$ and $q(x)$. It is easy to integrate $s(x)$. What about the $r(x)/q(x)$?</p>","metadata":{}},
{"cell_type":"markdown","source":"<p>We know from the <em>fundamental theorem of algebra</em> that we can factor $q(x) = p_1(x)^n_1 \\cdots p_k(x)^{n_k}$, where $p_i$ is a linear or quadratic factor. Further, the <a href=\"http://en.wikipedia.org/wiki/Partial_fraction_decomposition\">partial fraction decomposition</a> ensures then that the ratio can be written as:</p>","metadata":{}},
{"cell_type":"markdown","source":"\n$$\n\\frac{r(x)}{q(x)} = \\sum_i^k \\frac{a_{i1}(x)}{p_i(x)} + \\frac{a_{i2}(x)}{p_i(x)^2} + ... + \\frac{a_{in_i}(x)}{p_i(x)^{n_i}}\n$$\n","metadata":{}},
{"cell_type":"markdown","source":"<p>with the $a_{ij}(x)$ being polynomials of degree less than or equal the $p_i(x)$, so in this case either a constant or linear polynomial.</p>","metadata":{}},
{"cell_type":"markdown","source":"<p>So, if the polynomials of the form $(ax+b)/(cx^2 + dx + e)^j$ can be integrated, by the linearity of integration all rational functions can be integrated. Let's investigate this question.</p>","metadata":{}},
{"cell_type":"markdown","source":"<h3>Questions</h3>","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<ul>\n<li><p>The <code>apart</code> function separates rational functions into pieces. Use <code>apart</code> to find the partial fraction decomposition of</p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"\n$$\nf(x) = \\frac{1}{x(x+1)}\n$$\n","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<ul>\n<li><p>Use <code>apart</code> to find the partial fraction decomposition of </p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"\n$$\nf(x) = \\frac{1}{x(x+1)^5}\n$$\n","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<ul>\n<li><p>Use <code>apart</code> to find the partial fraction decomposition of </p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"\n$$\nf(x) = \\frac{1}{x(x^2+1)}\n$$\n","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<ul>\n<li><p>Use <code>apart</code> to find the partial fraction decomposition of </p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"\n$$\nf(x) = \\frac{1}{x(x^2+1)^3}\n$$\n","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<ul>\n<li><p>Use <code>apart</code> to find the partial fraction decomposition of </p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"\n$$\nf(x) = \\frac{1}{(x^2+2)^2 (x^2+3)^3}\n$$\n","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<ul>\n<li><p>Let $m$ be a positive integer. For which values of $m$ will </p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"\n$$\n\\int \\frac{1}{x^m} dx\n$$\n","metadata":{}},
{"cell_type":"markdown","source":"<p>will have a rational function for an answer? What is the answer when it isn't a rational function?</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<ul>\n<li><p>What is the integral of a term like:</p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"\n$$\n\\int \\frac{1}{x^2 + 1} dx\n$$\n","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<ul>\n<li><p>What is the integral of a term like</p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"\n$$\n\\int \\frac{x}{x^2 + 1} dx\n$$\n","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<ul>\n<li><p>For non-negative, integer values of $m$, do these integrals appear to always be rational functions? (Just try some different values starting with $m=1$.)</p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"\n$$\n\\int \\frac{1}{(x^2+1)^m} dx\n$$\n","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<ul>\n<li><p>For non-negative, integer values of $m$, do these integrals appear to always be rational functions?</p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"\n$$\n\\int \\frac{x}{(x^2+1)^m} dx\n$$\n","metadata":{}},
{"cell_type":"markdown","source":"<p>Verify this is the case for any $m$ by integrating the above symbolically. (You might also try to do the substitution manually.)</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1}
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   "file_extension": ".jl",                                                                                                             
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