{ "hash": "00e65a8e8895b92263e299d8d429a885", "result": { "markdown": "# Fundamental Theorem or Calculus\n\n\n\nThis section uses these add-on packages:\n\n``` {.julia .cell-code}\nusing CalculusWithJulia\nusing Plots\nusing SymPy\nusing Roots\nusing QuadGK\n```\n\n\n\n\n---\n\n\nWe refer to the example from the section on [transformations](../precalc/transformations.html#two_operators_D_S) where two operators on functions were defined:\n\n\n\n$$\nD(f)(k) = f(k) - f(k-1), \\quad S(f)(k) = f(1) + f(2) + \\cdots + f(k).\n$$\n\n\nIt was remarked that these relationships hold: $D(S(f))(k) = f(k)$ and $S(D(f))(k) = f(k) - f(0)$. These being a consequence of the inverse relationship between addition and subtraction. These two relationships are examples of a more general pair of relationships known as the [Fundamental theorem of calculus](http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus) or FTC.\n\n\nWe will see that with suitable rewriting, the derivative of a function is related to a certain limit of `D(f)` and the definite integral of a function is related to a certain limit of `S(f)`. The addition and subtraction rules encapsulated in the relations of $D(S(f))(k) = f(k)$ and $S(D(f))(k) = f(k) - f(0)$ then generalize to these calculus counterparts.\n\n\nThe FTC details the interconnectivity between the operations of integration and differentiation.\n\n\nFor example:\n\n\n> What is the definite integral of the derivative?\n\n\n\nThat is, what is $A = \\int_a^b f'(x) dx$? (Assume $f'$ is continuous.)\n\n\nTo investigate, we begin with the right Riemann sum using $h = (b-a)/n$:\n\n\n\n$$\nA \\approx S_n = \\sum_{i=1}^n f'(a + ih) \\cdot h.\n$$\n\n\nBut the mean value theorem says that for small $h$ we have $f'(x) \\approx (f(x) - f(x-h))/h$. Using this approximation with $x=a+ih$ gives:\n\n\n\n$$\nA \\approx\n\\sum_{i=1}^n \\left(f(a + ih) - f(a + (i-1)h)\\right).\n$$\n\n\nIf we let $g(i) = f(a + ih)$, then the summand above is just $g(i) - g(i-1) = D(g)(i)$ and the above then is just the sum of the $D(g)(i)$s, or:\n\n\n\n$$\nA \\approx S(D(g))(n) = g(n) - g(0).\n$$\n\n\nBut $g(n) - g(0) = f(a + nh) - f(a + 0h) = f(b) - f(a)$. That is, we expect that the $\\approx$ in the limit becomes $=$, or:\n\n\n\n$$\n\\int_a^b f'(x) dx = f(b) - f(a).\n$$\n\n\nThis is indeed the case.\n\n\nThe other question would be\n\n\n> What is the derivative of the integral?\n\n\n\nThat is, can we find the derivative of $\\int_0^x f(u) du$? (The derivative in $x$, the variable $u$ is a dummy variable of integration.)\n\n\nLet's look first at the integral using the right-Riemann sum, again using $h=(b-a)/n$:\n\n\n\n$$\n\\int_a^b f(u) du \\approx f(a + 1h)h + f(a + 2h)h + \\cdots f(a +nh)h = S(g)(n),\n$$\n\n\nwhere we define $g(i) = f(a + ih)h$. In the above, $n$ relates to $b$, but we could have stopped accumulating at any value. The analog for $S(g)(k)$ would be $\\int_a^x f(u) du$ where $x = a + kh$. That is we can make a function out of integration by considering the mapping $(x, \\int_a^x f(u) du)$. This might be written as $F(x) = \\int_a^x f(u)du$. With this definition, can we take a derivative in $x$?\n\n\nAgain, we fix a large $n$ and let $h=(b-a)/n$. And suppose $x = a + Mh$ for some $M$. Then writing out the approximations to both the definite integral and the derivative we have\n\n\n\n$$\n\\begin{align*}\nF'(x) = & \\frac{d}{dx} \\int_a^x f(u) du \\\\\n& \\approx \\frac{F(x) - F(x-h)}{h} \\\\\n&= \\frac{\\int_a^x f(u) du - \\int_a^{x-h} f(u) du}{h}\\\\\n& \\approx \\frac{\\left(f(a + 1h)h + f(a + 2h)h + \\cdots + f(a + (M-1)h)h + f(a + Mh)h\\right)}{h}\\\\\n&- \\quad\n\\frac{\\left(f(a + 1h)h + f(a + 2h)h + \\cdots + f(a + (M-1)h)h \\right)}{h} \\\\\n& = \\left(f(a + 1h) + \\quad f(a + 2h) + \\cdots + f(a + (M-1)h) + f(a + Mh)\\right)\\\\\n&- \\quad\n\\left(f(a + 1h) + f(a + 2h) + \\cdots + f(a + (M-1)h) \\right) \\\\\n&= f(a + Mh).\n\\end{align*}\n$$\n\n\nIf $g(i) = f(a + ih)$, then the above becomes\n\n\n\n$$\n\\begin{align*}\nF'(x) & \\approx D(S(g))(M) \\\\\n&= f(a + Mh)\\\\\n&= f(x).\n\\end{align*}\n$$\n\n\nThat is $F'(x) \\approx f(x)$.\n\n\nIn the limit, then, we would expect that\n\n\n\n$$\n\\frac{d}{dx} \\int_a^x f(u) du = f(x).\n$$\n\n\nWith these heuristics, we now have:\n\n\n> **The fundamental theorem of calculus**\n>\n> Part 1: Let $f$ be a continuous function on a closed interval $[a,b]$ and define $F(x) = \\int_a^x f(u) du$ for $a \\leq x \\leq b$. Then $F$ is continuous on $[a,b]$, differentiable on $(a,b)$ and moreover, $F'(x) =f(x)$.\n>\n> Part 2: Now suppose $f$ is any integrable function on a closed interval $[a,b]$ and $F(x)$ is *any* differentiable function on $[a,b]$ with $F'(x) = f(x)$. Then $\\int_a^b f(x)dx=F(b)-F(a)$.\n\n\n\n:::{.callout-note}\n## Note\nIn Part 1, the integral $F(x) = \\int_a^x f(u) du$ is defined for any Riemann integrable function, $f$. If the function is not continuous, then it is true the $F$ will be continuous, but it need not be true that it is differentiable at all points in $(a,b)$. Forming $F$ from $f$ is a form of *smoothing*. It makes a continuous function out of an integrable one, a differentiable function from a continuous one, and a $k+1$-times differentiable function from a $k$-times differentiable one.\n\n:::\n\n## Using the fundamental theorem of calculus to evaluate definite integrals\n\n\nThe major use of the FTC is the computation of $\\int_a^b f(x) dx$. Rather then resort to Riemann sums or geometric arguments, there is an alternative - *when possible*, find a function $F$ with $F'(x) = f(x)$ and compute $F(b) - F(a)$.\n\n\nSome examples:\n\n\n * Consider the problem of Archimedes, $\\int_0^1 x^2 dx$. Clearly, we have with $f(x) = x^2$ that $F(x)=x^3/3$ will satisfy the assumptions of the FTC, so that:\n\n\n\n$$\n\\int_0^1 x^2 dx = F(1) - F(0) = \\frac{1^3}{3} - \\frac{0^3}{3} = \\frac{1}{3}.\n$$\n\n\n * More generally, we know if $n\\neq-1$ that if $f(x) = x^{n}$, that\n\n\n\n$$\nF(x) = x^{n+1}/(n+1)\n$$\n\n\nwill satisfy $F'(x)=f(x)$, so that\n\n\n\n$$\n\\int_a^b x^n dx = \\frac{b^{n+1} - a^{n+1}}{n+1}, \\quad n\\neq -1.\n$$\n\n\n(Well almost! We must be careful to know that $a \\cdot b > 0$, as otherwise we will encounter a place where $f(x)$ may not be integrable.)\n\n\nWe note that the above includes the case of a constant, or $n=0$.\n\n\nWhat about the case $n=-1$, or $f(x) = 1/x$, that is not covered by the above? For this special case, it is known that $F(x) = \\log(x)$ (natural log) will have $F'(x) = 1/x$. This gives for $0 < a < b$:\n\n\n\n$$\n\\int_a^b \\frac{1}{x} dx = \\log(b) - \\log(a).\n$$\n\n\n * Let $f(x) = \\cos(x)$. How much area is between $-\\pi/2$ and $\\pi/2$? We have that $F(x) = \\sin(x)$ will have $F'(x) = f(x)$, so:\n\n\n\n$$\n\\int_{-\\pi/2}^{\\pi/2} \\cos(x) dx = F(\\pi/2) - F(-\\pi/2) = 1 - (-1) = 2.\n$$\n\n\n### An alternate notation for $F(b) - F(a)$\n\n\nThe expression $F(b) - F(a)$ is often written in this more compact form:\n\n\n\n$$\n\\int_a^b f(x) dx = F(b) - F(a) = F(x)\\big|_{x=a}^b, \\text{ or just expr}\\big|_{x=a}^b.\n$$\n\n\nThe vertical bar is used for the *evaluation* step, in this case the $a$ and $b$ mirror that of the definite integral. This notation lends itself to working inline, as we illustrate with this next problem where we \"know\" a function \"$F$\", so just express it \"inline\":\n\n\n\n$$\n\\int_0^{\\pi/4} \\sec^2(x) dx = \\tan(x) \\big|_{x=0}^{\\pi/4} = 1 - 0 = 1.\n$$\n\n\nA consequence of this notation is:\n\n\n\n$$\nF(x) \\big|_{x=a}^b = -F(x) \\big|_{x=b}^a.\n$$\n\n\nThis says nothing more than $F(b)-F(a) = -F(a) - (-F(b))$, though more compactly.\n\n\n## The indefinite integral\n\n\nA function $F(x)$ with $F'(x) = f(x)$ is known as an *antiderivative* of $f$. For a given $f$, there are infinitely many antiderivatives: if $F(x)$ is one, then so is $G(x) = F(x) + C$. But - due to the mean value theorem - all antiderivatives for $f$ differ at most by a constant.\n\n\nThe **indefinite integral** of $f(x)$ is denoted by:\n\n\n\n$$\n\\int f(x) dx.\n$$\n\n\n(There are no limits of integration.) There are two possible definitions: this refers to the set of *all* antiderivatives, or is just one of the set of all antiderivatives for $f$. The former gives rise to expressions such as\n\n\n\n$$\n\\int x^2 dx = \\frac{x^3}{3} + C\n$$\n\n\nwhere $C$ is the *constant of integration* and isn't really a fixed constant, but any possible constant. These notes will follow the lead of `SymPy` and not give a $C$ in the expression, but instead rely on the reader to understand that there could be many other possible expressions given, though all differ by no more than a constant. This means, that $\\int f(x) dx$ refers to *an* antiderivative, not *the* collection of all antiderivatives.\n\n\n### The `integrate` function from `SymPy`\n\n\n`SymPy` provides the `integrate` function to perform integration. There are two usages:\n\n\n * `integrate(ex, var)` to find an antiderivative\n * `integrate(ex, (var, a, b))` to find the definite integral. This integrates the expression in the variable `var` from `a` to `b`.\n\n\nTo illustrate, we have, this call finds an antiderivative:\n\n::: {.cell execution_count=4}\n``` {.julia .cell-code}\n@syms x\nintegrate(sin(x),x)\n```\n\n::: {.cell-output .cell-output-display execution_count=5}\n```{=html}\n \n\\[\n- \\cos{\\left(x \\right)}\n\\]\n\n```\n:::\n:::\n\n\nWhereas this call computes the \"area\" under $f(x)$ between `a` and `b`:\n\n::: {.cell execution_count=5}\n``` {.julia .cell-code}\nintegrate(sin(x), (x, 0, pi))\n```\n\n::: {.cell-output .cell-output-display execution_count=6}\n```{=html}\n \n\\[\n2.0\n\\]\n\n```\n:::\n:::\n\n\nAs does this for a different function:\n\n::: {.cell execution_count=6}\n``` {.julia .cell-code}\nintegrate(acos(1-x), (x, 0, 2))\n```\n\n::: {.cell-output .cell-output-display execution_count=7}\n```{=html}\n \n\\[\n\\pi\n\\]\n\n```\n:::\n:::\n\n\nAnswers may depend on conditions, as here, where the case $n=-1$ breaks a pattern:\n\n::: {.cell hold='true' execution_count=7}\n``` {.julia .cell-code}\n@syms x::real n::real\nintegrate(x^n, x) # indefinite integral\n```\n\n::: {.cell-output .cell-output-display execution_count=8}\n```{=html}\n \n\\[\n\\begin{cases} \\frac{x^{n + 1}}{n + 1} & \\text{for}\\: n \\neq -1 \\\\\\log{\\left(x \\right)} & \\text{otherwise} \\end{cases}\n\\]\n\n```\n:::\n:::\n\n\nAnswers may depend on specific assumptions:\n\n::: {.cell hold='true' execution_count=8}\n``` {.julia .cell-code}\n@syms u\nintegrate(abs(u),u)\n```\n\n::: {.cell-output .cell-output-display execution_count=9}\n```{=html}\n \n\\[\n\\int \\left|{u}\\right|\\, du\n\\]\n\n```\n:::\n:::\n\n\nYet\n\n::: {.cell hold='true' execution_count=9}\n``` {.julia .cell-code}\n@syms u::real\nintegrate(abs(u),u)\n```\n\n::: {.cell-output .cell-output-display execution_count=10}\n```{=html}\n \n\\[\n\\begin{cases} - \\frac{u^{2}}{2} & \\text{for}\\: u \\leq 0 \\\\\\frac{u^{2}}{2} & \\text{otherwise} \\end{cases}\n\\]\n\n```\n:::\n:::\n\n\nAnswers may not be available as elementary functions, but there may be special functions that have special cases.\n\n::: {.cell hold='true' execution_count=10}\n``` {.julia .cell-code}\n@syms x::real\nintegrate(x / sqrt(1-x^3), x)\n```\n\n::: {.cell-output .cell-output-display execution_count=11}\n```{=html}\n \n\\[\n\\frac{x^{2} \\Gamma\\left(\\frac{2}{3}\\right) {{}_{2}F_{1}\\left(\\begin{matrix} \\frac{1}{2}, \\frac{2}{3} \\\\ \\frac{5}{3} \\end{matrix}\\middle| {x^{3} e^{2 i \\pi}} \\right)}}{3 \\Gamma\\left(\\frac{5}{3}\\right)}\n\\]\n\n```\n:::\n:::\n\n\nThe different cases explored by `integrate` are after the questions.\n\n\n## Rules of integration\n\n\nThere are some \"rules\" of integration that allow integrals to be re-expressed. These follow from the rules of derivatives.\n\n\n * The integral of a constant times a function:\n\n\n\n$$\n\\int c \\cdot f(x) dx = c \\cdot \\int f(x) dx.\n$$\n\n\nThis follows as if $F(x)$ is an antiderivative of $f(x)$, then $[cF(x)]' = c f(x)$ by rules of derivatives.\n\n\n * The integral of a sum of functions:\n\n\n\n$$\n\\int (f(x) + g(x)) dx = \\int f(x) dx + \\int g(x) dx.\n$$\n\n\nThis follows immediately as if $F(x)$ and $G(x)$ are antiderivatives of $f(x)$ and $g(x)$, then $[F(x) + G(x)]' = f(x) + g(x)$, so the right hand side will have a derivative of $f(x) + g(x)$.\n\n\nIn fact, this more general form where $c$ and $d$ are constants covers both cases:\n\n\n\n$$\n\\int (cf(x) + dg(x)) dx = c \\int f(x) dx + d \\int g(x) dx.\n$$\n\n\nThis statement is nothing more than the derivative formula $[cf(x) + dg(x)]' = cf'(x) + dg'(x)$. The product rule gives rise to a technique called *integration by parts* and the chain rule gives rise to a technique of *integration by substitution*, but we defer those discussions to other sections.\n\n\n##### Examples\n\n\n * The antiderivative of the polynomial $p(x) = a_n x^n + \\cdots a_1 x + a_0$ follows from the linearity of the integral and the general power rule:\n\n\n\n$$\n\\begin{align}\n\\int (a_n x^n + \\cdots a_1 x + a_0) dx\n&= \\int a_nx^n dx + \\cdots \\int a_1 x dx + \\int a_0 dx \\\\\n&= a_n \\int x^n dx + \\cdots + a_1 \\int x dx + a_0 \\int dx \\\\\n&= a_n\\frac{x^{n+1}}{n+1} + \\cdots + a_1 \\frac{x^2}{2} + a_0 \\frac{x}{1}.\n\\end{align}\n$$\n\n\n * More generally, a [Laurent](https://en.wikipedia.org/wiki/Laurent_polynomial) polynomial allows for terms with negative powers. These too can be handled by the above. For example\n\n\n\n$$\n\\begin{align}\n\\int (\\frac{2}{x} + 2 + 2x) dx\n&= \\int \\frac{2}{x} dx + \\int 2 dx + \\int 2x dx \\\\\n&= 2\\int \\frac{1}{x} dx + 2 \\int dx + 2 \\int xdx\\\\\n&= 2\\log(x) + 2x + 2\\frac{x^2}{2}.\n\\end{align}\n$$\n\n\n * Consider this integral:\n\n\n\n$$\n\\int_0^\\pi 100 \\sin(x) dx = F(\\pi) - F(0),\n$$\n\n\nwhere $F(x)$ is an antiderivative of $100\\sin(x)$. But:\n\n\n\n$$\n\\int 100 \\sin(x) dx = 100 \\int \\sin(x) dx = 100 (-\\cos(x)).\n$$\n\n\nSo the answer to the question is\n\n\n\n$$\n\\int_0^\\pi 100 \\sin(x) dx = (100 (-\\cos(\\pi))) - (100(-\\cos(0))) = (100(-(-1))) - (100(-1)) = 200.\n$$\n\n\nThis seems like a lot of work, and indeed it is more than is needed. The following would be more typical once the rules are learned:\n\n\n\n$$\n\\int_0^\\pi 100 \\sin(x) dx = 100(-\\cos(x)) \\big|_0^{\\pi} = 100 \\cos(x) \\big|_{\\pi}^0 = 100(1) - 100(-1) = 200.\n$$\n\n\n## The derivative of the integral\n\n\nThe relationship that $[\\int_a^x f(u) du]' = f(x)$ is a bit harder to appreciate, as it doesn't help answer many ready made questions. Here we give some examples of its use.\n\n\nFirst, the expression defining an antiderivative, or indefinite integral, is given in term of a definite integral:\n\n\n\n$$\nF(x) = \\int_a^x f(u) du.\n$$\n\n\nThe value of $a$ does not matter, as long as the integral is defined.\n\n::: {.cell cache='true' hold='true' execution_count=11}\n\n::: {.cell-output .cell-output-display execution_count=12}\n```{=html}\n
\"A\n

Illustration showing \\(F(x) = \\int_a^x f(u) du\\) is a function that accumulates area. The value of \\(A\\) is the area over \\([x_{n-1}, x_n]\\) and also the difference \\(F(x_n) - F(x_{n-1})\\).

\n
\n
\n
\n```\n:::\n:::\n\n\nThe picture for this, for non-negative $f$, is of accumulating area as $x$ increases. It can be used to give insight into some formulas:\n\n\nFor any function, we know that $F(b) - F(c) + F(c) - F(a) = F(b) - F(a)$. For this specific function, this translates into this property of the integral:\n\n\n\n$$\n\\int_a^b f(x) dx = \\int_a^c f(x) dx + \\int_c^b f(x) dx.\n$$\n\n\nSimilarly, $\\int_a^a f(x) dx = F(a) - F(a) = 0$ follows.\n\n\nTo see that the value of $a$ does not matter, consider $a_0 < a_1$. Then we have with\n\n\n\n$$\nF(x) = \\int_{a_0}^x f(u)du, \\quad G(x) = \\int_{a_0}^x f(u)du,\n$$\n\n\nThat $F(x) = G(x) + \\int_{a_0}^{a_1} f(u) du$. The additional part may look complicated, but the point is that as far as $x$ is involved, it is a constant. Hence both $F$ and $G$ are antiderivatives if either one is.\n\n\n##### Example\n\n\nFrom the familiar formula rate $\\times$ time $=$ distance, we \"know,\" for example, that a car traveling 60 miles an hour for one hour will have traveled 60 miles. This allows us to translate statements about the speed (or more generally velocity) into statements about position at a given time. If the speed is not constant, we don't have such an easy conversion.\n\n\nSuppose our velocity at time $t$ is $v(t)$, and always positive. We want to find the position at time $t$, $x(t)$. Let's assume $x(0) = 0$. Let $h$ be some small time step, say $h=(t - 0)/n$ for some large $n>0$. Then we can *approximate* $v(t)$ between $[ih, (i+1)h)$ by $v(ih)$. This is a constant so the change in position over the time interval $[ih, (i+1)h)$ would simply be $v(ih) \\cdot h$, and ignoring the accumulated errors, the approximate position at time $t$ would be found by adding this pieces together: $x(t) \\approx v(0h)\\cdot h + v(1h)\\cdot h + v(2h) \\cdot h + \\cdots + v(nh)h$. But we recognize this (as did [Beeckman](http://www.math.harvard.edu/~knill/teaching/math1a_2011/exhibits/bressoud/) in 1618) as nothing more than an approximation for the Riemann sum of $v$ over the interval $[0, t]$. That is, we expect:\n\n\n\n$$\nx(t) = \\int_0^t v(u) du.\n$$\n\n\nHopefully this makes sense: our position is the result of accumulating our change in position over small units of time. The old one-foot-in-front-of-another approach to walking out the door.\n\n\nThe above was simplified by the assumption that $x(0) = 0$. What if $x(0) = x_0$ for some non-zero value. Then the above is not exactly correct, as $\\int_0^0 v(u) du = 0$. So instead, we might write this more concretely as:\n\n\n\n$$\nx(t) = x_0 + \\int_0^t v(u) du.\n$$\n\n\nThere is a similar relationship between velocity and acceleration, but let's think about it formally. If we know that the acceleration is the rate of change of velocity, then we have $a(t) = v'(t)$. By the FTC, then\n\n\n\n$$\n\\int_0^t a(u) du = \\int_0^t v'(t) = v(t) - v(0).\n$$\n\n\nRewriting gives a similar statement as before:\n\n\n\n$$\nv(t) = v_0 + \\int_0^t a(u) du.\n$$\n\n\n##### Example\n\n\nIn probability theory, for a positive, continuous random variable, the probability that the random value is less than $a$ is given by $P(X \\leq a) = F(a) = \\int_{0}^a f(x) dx$. (Positive means the integral starts at $0$, whereas in general it could be $-\\infty$, a minor complication that we haven't yet discussed.)\n\n\nFor example, the exponential distribution with rate $1$ has $f(x) = e^{-x}$. Compute $F(x)$.\n\n\nThis is just $F(x) = \\int_0^x e^{-u} du = -e^{-u}\\big|_0^x = 1 - e^{-x}$.\n\n\nThe \"uniform\" distribution on $[a,b]$ has\n\n\n\n$$\nF(x) =\n\\begin{cases}\n0 & x < a\\\\\n\\frac{x-a}{b-a} & a \\leq x \\leq b\\\\\n1 & x > b\n\\end{cases}\n$$\n\n\nFind $f(x)$. There are some subtleties here. If we assume that $F(x) = \\int_0^x f(u) du$ then we know if $f(x)$ is continuous that $F'(x) = f(x)$. Differentiating we get\n\n\n\n$$\nf(x) = \\begin{cases}\n0 & x < a\\\\\n\\frac{1}{b-a} & a < x < b\\\\\n0 & x > b\n\\end{cases}\n$$\n\n\nHowever, the function $f$ is *not* continuous on $[a,b]$ and $F'(x)$ is not differentiable on $(a,b)$. It is true that $f$ is integrable, and where $F$ is differentiable $F'=f$. So $f$ is determined except possibly at the points $x=a$ and $x=b$.\n\n\n##### Example\n\n\nThe error function is defined by $\\text{erf}(x) = 2/\\sqrt{\\pi}\\int_0^x e^{-u^2} du$. It is implemented in `Julia` through `erf`. Suppose, we were to ask where it takes on it's maximum value, what would we find?\n\n\nThe answer will either be at a critical point, at $0$ or as $x$ goes to $\\infty$. We can differentiate to find critical points:\n\n\n\n$$\n[\\text{erf}(x)] = \\frac{2}{\\pi}e^{-x^2}.\n$$\n\n\nOh, this is never $0$, so there are no critical points. The maximum occurs at $0$ or as $x$ goes to $\\infty$. Clearly at $0$, we have $\\text{erf}(0)=0$, so the answer will be as $x$ goes to $\\infty$.\n\n\nIn retrospect, this is a silly question. As $f(x) > 0$ for all $x$, we *must* have that $F(x)$ is strictly increasing, so never gets to a local maximum.\n\n\n##### Example\n\n\nThe [Dawson](http://en.wikipedia.org/wiki/Dawson_function) function is\n\n\n\n$$\nF(x) = e^{-x^2} \\int_0^x e^{t^2} dt\n$$\n\n\nCharacterize any local maxima or minima.\n\n\nFor this we need to consider the product rule. The fundamental theorem of calculus will help with the right-hand side. We have:\n\n\n\n$$\nF'(x) = (-2x)e^{-x^2} \\int_0^x e^{t^2} dt + e^{-x^2} e^{x^2} = -2x F(x) + 1\n$$\n\n\nWe need to figure out when this is $0$. For that, we use some numeric math.\n\n::: {.cell execution_count=12}\n``` {.julia .cell-code}\nF(x) = exp(-x^2) * quadgk(t -> exp(t^2), 0, x)[1]\nFp(x) = -2x*F(x) + 1\ncps = find_zeros(Fp, -4, 4)\n```\n\n::: {.cell-output .cell-output-display execution_count=13}\n```\n2-element Vector{Float64}:\n -0.9241388730045916\n 0.9241388730045916\n```\n:::\n:::\n\n\nWe could take a second derivative to characterize. For that we use $F''(x) = [-2xF(x) + 1]' = -2F(x) + -2x(-2xF(x) + 1)$, so\n\n::: {.cell execution_count=13}\n``` {.julia .cell-code}\nFpp(x) = -2F(x) + 4x^2*F(x) - 2x\nFpp.(cps)\n```\n\n::: {.cell-output .cell-output-display execution_count=14}\n```\n2-element Vector{Float64}:\n 1.0820884492703637\n -1.0820884492703637\n```\n:::\n:::\n\n\nThe first value being positive says there is a relative minimum at $-0.924139$, at $0.924139$ there is a relative maximum.\n\n\n##### Example\n\n\nReturning to probability, suppose there are $n$ positive random numbers $X_1$, $X_2$, ..., $X_n$. A natural question might be to ask what formulas describes the largest of these values, assuming each is identical in some way. A description that is helpful is to define $F(a) = P(X \\leq a)$ for some random number $X$. That is the probability that $X$ is less than or equal to $a$ is $F(a)$. For many situations, there is a *density* function, $f$, for which $F(a) = \\int_0^a f(x) dx$.\n\n\nUnder assumptions that the $X$ are identical and independent, the largest value, $M$, may b characterized by $P(M \\leq a) = \\left[F(a)\\right]^n$. Using $f$ and $F$ describe the derivative of this expression.\n\n\nThis problem is constructed to take advantage of the FTC, and we have:\n\n\n\n$$\n\\begin{align*}\n\\left[P(M \\leq a)\\right]'\n&= \\left[F(a)^n\\right]'\\\\\n&= n \\cdot F(a)^{n-1} \\left[F(a)\\right]'\\\\\n&= n F(a)^{n-1}f(a)\n\\end{align*}\n$$\n\n\n##### Example\n\n\nSuppose again probabilities of a random number between $0$ and $1$, say, are given by a positive, continuous function $f(x)$on $(0,1)$ by $F(a) = P(X \\leq a) = \\int_0^a f(x) dx$. The median value of the random number is a value of $a$ for which $P(X \\leq a) = 1/2$. Such an $a$ makes $X$ a coin toss – betting if $X$ is less than $a$ is like betting on heads to come up. More generally the $q$th quantile of $X$ is a number $a$ with $P(X \\leq a) = q$. The definition is fine, but for a given $f$ and $q$ can we find $a$?\n\n\nAbstractly, we are solving $F(a) = q$ or $F(a)-q = 0$ for $a$. That is, this is a zero-finding question. We have discussed different options for this problem: bisection, a range of derivative free methods, and Newton's method. As evaluating $F$ involves an integral, which may involve many evaluations of $f$, a method which converges quickly is preferred. For that, Newton's method is a good idea, it having quadratic convergence in this case, as $a$ is a simple zero given that $F$ is increasing under the assumptions above.\n\n\nNewton's method involves the update step `x = x - f(x)/f'(x)`. For this \"$f$\" is $h(x) = \\int_0^x f(u) du - q$. The derivative is easy, the FTC just applies: $h'(x) = f(x)$; no need for automatic differentiation, which may not even apply to this setup.\n\n\nTo do a concrete example, we take the [Beta](https://en.wikipedia.org/wiki/Beta_distribution)($\\alpha, \\beta$) distribution ($\\alpha, \\beta > 0$) which has density, $f$, over $[0,1]$ given by\n\n\n\n$$\nf(x) = x^{\\alpha-1}\\cdot (1-x)^{\\beta-1} \\cdot \\frac{\\Gamma(\\alpha+\\beta)}{\\Gamma(\\alpha)\\Gamma(\\beta)}\n$$\n\n\nThe Wikipedia link above gives an approximate answer for the median of $(\\alpha-1/3)/(\\alpha+\\beta-2/3)$ when $\\alpha,\\beta > 1$. Let's see how correct this is when $\\alpha=5$ and $\\beta=6$. The `gamma` function used below implements $\\Gamma$. It is in the `SpecialFunctions` package, which is loaded with the `CalculusWithJulia` package.\n\n::: {.cell execution_count=14}\n``` {.julia .cell-code}\nalpha, beta = 5,6\nf(x) = x^(alpha-1)*(1-x)^(beta-1) * gamma(alpha + beta) / (gamma(alpha) * gamma(beta))\nq = 1/2\nh(x) = first(quadgk(f, 0, x)) - q\nhp(x) = f(x)\n\nx0 = (alpha-1/3)/(alpha + beta - 2/3)\nxstar = find_zero((h, hp), x0, Roots.Newton())\n\nxstar, x0\n```\n\n::: {.cell-output .cell-output-display execution_count=15}\n```\n(0.4516941562236631, 0.45161290322580644)\n```\n:::\n:::\n\n\nThe asymptotic answer agrees with the answer in the first four decimal places.\n\n\nAs an aside, we ask how many function evaluations were taken? We can track this with a trick - using a closure to record when $f$ is called:\n\n::: {.cell execution_count=15}\n``` {.julia .cell-code}\nfunction FnWrapper(f)\n ctr = 0\n function(x)\n ctr += 1\n f(x)\n end\nend\n```\n\n::: {.cell-output .cell-output-display execution_count=16}\n```\nFnWrapper (generic function with 1 method)\n```\n:::\n:::\n\n\nThen we have the above using `FnWrapper(f)` in place of `f`:\n\n::: {.cell hold='true' execution_count=16}\n``` {.julia .cell-code}\nff = FnWrapper(f)\nF(x) = first(quadgk(ff, 0, x))\nh(x) = F(x) - q\nhp(x) = ff(x)\nxstar = find_zero((h, hp), x0, Roots.Newton())\nxstar, ff.ctr\n```\n\n::: {.cell-output .cell-output-display execution_count=17}\n```\n(0.4516941562236631, Core.Box(48))\n```\n:::\n:::\n\n\nSo the answer is the same. Newton's method converged in 3 steps, and called `h` or `hp` 5 times.\n\n\nAssuming the number inside `Core.Box` is the value of `ctr`, we see not so many function calls, just $48$.\n\n\nWere `f` very expensive to compute or `h` expensive to compute (which can happen if, say, `f` were highly oscillatory) then steps could be made to cut this number down, such as evaluating $F(x_n) = \\int_{x_0}^{x_n} f(x) dx$, using linearity, as $\\int_0^{x_0} f(x) dx + \\int_{x_0}^{x_1}f(x)dx + \\int_{x_1}^{x_2}f(x)dx + \\cdots + \\int_{x_{n-1}}^{x_n}f(x)dx$. Then all but the last term could be stored from the previous steps of Newton's method. The last term presumably being less costly as it would typically involve a small interval.\n\n\n:::{.callout-note}\n## Note\nThe trick using a closure relies on an internal way of accessing elements in a closure. The same trick could be implemented many different ways which aren't reliant on undocumented internals, this approach was just a tad more convenient. It shouldn't be copied for work intended for distribution, as the internals may change without notice or deprecation.\n\n:::\n\n##### Example\n\n\nA junior engineer at `Treadmillz.com` is tasked with updating the display of calories burned for an older-model treadmill. The old display involved a sequence of LED \"dots\" that updated each minute. The last 10 minutes were displayed. Each dot corresponded to one calorie burned, so the total number of calories burned in the past 10 minutes was the number of dots displayed, or the sum of each column of dots. An example might be:\n\n``` {.julia .cell-code}\n **\n ****\n *****\n ********\n**********\n```\n\n\nIn this example display there was 1 calorie burned in the first minute, then 2, then 5, 5, 4, 3, 2, 2, 1. The total is $24$.\n\n\nIn her work the junior engineer found this old function for updating the display\n\n``` {.julia .cell-code}\nfunction cnew = update(Cnew, Cold)\n cnew = Cnew - Cold\nend\n```\n\n\nShe discovered that the function was written awhile ago, and in MATLAB. The function receives the values `Cnew` and `Cold` which indicate the *total* number of calories burned up until that time frame. The value `cnew` is the number of calories burned in the minute. (Some other engineer has cleverly figured out how many calories have been burned during the time on the machine.)\n\n\nThe new display will have twice as many dots, so the display can be updated every 30 seconds and still display 10 minutes worth of data. What should the `update` function now look like?\n\n\nHer first attempt was simply to rewrite the function in `Julia`:\n\n::: {.cell execution_count=19}\n``` {.julia .cell-code}\nfunction update₁(Cnew, Cold)\n cnew = Cnew - Cold\nend\n```\n\n::: {.cell-output .cell-output-display execution_count=18}\n```\nupdate₁ (generic function with 1 method)\n```\n:::\n:::\n\n\nThis has the advantage that each \"dot\" still represents a calorie burned, so that a user can still count the dots to see the total burned in the past 10 minutes.\n\n``` {.julia .cell-code}\n * *\n ****** *\n ************* *\n```\n\n\nSadly though, users didn't like it. Instead of a set of dots being, say, 5 high, they were now 3 high and 2 high. It \"looked\" like they were doing less work! What to do?\n\n\nThe users actually were not responding to the number of dots, which hadn't changed, but rather the *area* that they represented - and this shrank in half. (It is much easier to visualize area than count dots when tired.) How to adjust for that?\n\n\nWell our engineer knew - double the dots and count each as half a calorie. This makes the \"area\" constant. She also generalized letting `n` be the number of updates per minute, in anticipation of even further improvements in the display technology:\n\n::: {.cell execution_count=21}\n``` {.julia .cell-code}\nfunction update(Cnew, Cold, n)\n cnew = (Cnew - Cold) * n\nend\n```\n\n::: {.cell-output .cell-output-display execution_count=19}\n```\nupdate (generic function with 1 method)\n```\n:::\n:::\n\n\nThen the \"area\" represented by the dots stays fixed over this time frame.\n\n\nThe engineer then thought a bit more, as the form of her answer seemed familiar. She decides to parameterize it in terms of $t$ and found with $h=1/n$: `c(t) = (C(t) - C(t-h))/h`. Ahh - the derivative approximation. But then what is the \"area\"? It is no longer just the sum of the dots, but in terms of the functions she finds that each column represents $c(t)\\cdot h$, and the sum is just $c(t_1)h + c(t_2)h + \\cdots c(t_n)h$ which looks like an approximate integral.\n\n\nIf the display were to reach the modern age and replace LED \"dots\" with a higher-pixel display, then the function to display would be $c(t) = C'(t)$ and the area displayed would be $\\int_{t-10}^t c(u) du$.\n\n\nThinking a bit harder, she knows that her `update` function is getting $C(t)$, and displaying the *rate* of calorie burn leads to the area displayed being interpretable as the total calories burned between $t$ and $t-10$ (or $C(t)-C(t-10)$) by the fundamental theorem of calculus.\n\n\n## Questions\n\n\n###### Question\n\n\nIf $F(x) = e^{x^2}$ is an antiderivative for $f$, find $\\int_0^2 f(x) dx$.\n\n::: {.cell hold='true' execution_count=22}\n\n::: {.cell-output .cell-output-display execution_count=20}\n```{=html}\n
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\n\n\n```\n:::\n:::\n\n\n###### Question\n\n\nIf $\\sin(x) - x\\cos(x)$ is an antiderivative for $x\\sin(x)$, find the following integral $\\int_0^\\pi x\\sin(x) dx$.\n\n::: {.cell hold='true' execution_count=23}\n\n::: {.cell-output .cell-output-display execution_count=21}\n```{=html}\n
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\n\n\n```\n:::\n:::\n\n\n###### Question\n\n\nFind an antiderivative then evaluate $\\int_0^1 x(1-x) dx$.\n\n::: {.cell hold='true' execution_count=24}\n\n::: {.cell-output .cell-output-display execution_count=22}\n```{=html}\n
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\n\n\n```\n:::\n:::\n\n\n###### Question\n\n\nUse the fact that $[e^x]' = e^x$ to evaluate $\\int_0^e (e^x - 1) dx$.\n\n::: {.cell hold='true' execution_count=25}\n\n::: {.cell-output .cell-output-display execution_count=23}\n```{=html}\n
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\n\n\n```\n:::\n:::\n\n\n###### Question\n\n\nFind the value of $\\int_0^1 (1-x^2/2 + x^4/24) dx$.\n\n::: {.cell hold='true' execution_count=26}\n\n::: {.cell-output .cell-output-display execution_count=24}\n```{=html}\n
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\n\n\n```\n:::\n:::\n\n\n###### Question\n\n\nUsing `SymPy`, what is an antiderivative for $x^2 \\sin(x)$?\n\n::: {.cell hold='true' execution_count=27}\n\n::: {.cell-output .cell-output-display execution_count=25}\n```{=html}\n
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\n\n\n```\n:::\n:::\n\n\n###### Question\n\n\nUsing `SymPy`, what is an antiderivative for $xe^{-x}$?\n\n::: {.cell hold='true' execution_count=28}\n\n::: {.cell-output .cell-output-display execution_count=26}\n```{=html}\n
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\n\n\n```\n:::\n:::\n\n\n###### Question\n\n\nUsing `SymPy`, integrate the function $\\int_0^{2\\pi} e^x \\cdot \\sin(x) dx$.\n\n::: {.cell hold='true' execution_count=29}\n\n::: {.cell-output .cell-output-display execution_count=27}\n```{=html}\n
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\n\n\n```\n:::\n:::\n\n\n###### Question\n\n\nA particle has velocity $v(t) = 2t^2 - t$ between $0$ and $1$. If $x(0) = 0$, find the position $x(1)$.\n\n::: {.cell hold='true' execution_count=30}\n\n::: {.cell-output .cell-output-display execution_count=28}\n```{=html}\n
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\n\n\n```\n:::\n:::\n\n\n###### Question\n\n\nA particle has acceleration given by $\\sin(t)$ between $0$ and $\\pi$. If the initial velocity is $v(0) = 0$, find $v(\\pi/2)$.\n\n::: {.cell hold='true' execution_count=31}\n\n::: {.cell-output .cell-output-display execution_count=29}\n```{=html}\n
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\n\n\n```\n:::\n:::\n\n\n###### Question\n\n\nThe position of a particle is given by $x(t) = \\int_0^t g(u) du$, where $x(0)=0$ and $g(u)$ is given by this piecewise linear graph:\n\n::: {.cell hold='true' execution_count=32}\n\n::: {.cell-output .cell-output-display execution_count=30}\n![](ftc_files/figure-html/cell-33-output-1.svg){}\n:::\n:::\n\n\n * The velocity of the particle is positive over:\n\n::: {.cell hold='true' execution_count=33}\n\n::: {.cell-output .cell-output-display execution_count=31}\n```{=html}\n
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\n\n\n```\n:::\n:::\n\n\n * The position of the particle is $0$ at $t=0$ and:\n\n::: {.cell hold='true' execution_count=34}\n\n::: {.cell-output .cell-output-display execution_count=32}\n```{=html}\n
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\n\n\n```\n:::\n:::\n\n\n * The position of the particle at time $t=5$ is?\n\n::: {.cell hold='true' execution_count=35}\n\n::: {.cell-output .cell-output-display execution_count=33}\n```{=html}\n
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\n\n\n```\n:::\n:::\n\n\n * On the interval $[2,3]$:\n\n::: {.cell hold='true' execution_count=36}\n\n::: {.cell-output .cell-output-display execution_count=34}\n```{=html}\n
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\n\n\n```\n:::\n:::\n\n\n###### Question\n\n\nLet $F(x) = \\int_{t-10}^t f(u) du$ for $f(u)$ a positive, continuous function. What is $F'(t)$?\n\n::: {.cell hold='true' execution_count=37}\n\n::: {.cell-output .cell-output-display execution_count=35}\n```{=html}\n
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\n\n\n```\n:::\n:::\n\n\n###### Question\n\n\nSuppose $f(x) \\geq 0$ and $F(x) = \\int_0^x f(u) du$. $F(x)$ is continuous and so has a maximum value on the interval $[0,1]$ taken at some $c$ in $[0,1]$. It is\n\n::: {.cell hold='true' execution_count=38}\n\n::: {.cell-output .cell-output-display execution_count=36}\n```{=html}\n
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\n\n\n```\n:::\n:::\n\n\n###### Question\n\n\nLet $F(x) = \\int_0^x f(u) du$, where $f(x)$ is given by the graph below. Identify the $x$ values of all *relative maxima* of $F(x)$. Explain why you know these are the values.\n\n::: {.cell hold='true' execution_count=39}\n\n::: {.cell-output .cell-output-display execution_count=37}\n![](ftc_files/figure-html/cell-40-output-1.svg){}\n:::\n:::\n\n\n::: {.cell hold='true' execution_count=40}\n\n::: {.cell-output .cell-output-display execution_count=38}\n```{=html}\n
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\n\n\n```\n:::\n:::\n\n\n###### Question\n\n\nSuppose $f(x)$ is monotonically decreasing with $f(0)=1$, $f(1/2) = 0$ and $f(1) = -1$. Let $F(x) = \\int_0^x f(u) du$. $F(x)$ is continuous and so has a maximum value on the interval $[0,1]$ taken at some $c$ in $[0,1]$. It is\n\n::: {.cell hold='true' execution_count=41}\n\n::: {.cell-output .cell-output-display execution_count=39}\n```{=html}\n
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\n\n\n```\n:::\n:::\n\n\n###### Question\n\n\nBarrow presented a version of the fundamental theorem of calculus in a 1670 volume edited by Newton, Barrow's student (cf. [Wagner](http://www.maa.org/sites/default/files/0746834234133.di020795.02p0640b.pdf)). His version can be stated as follows (cf. [Jardine](http://www.maa.org/publications/ebooks/mathematical-time-capsules)):\n\n\nConsider the following figure where $f$ is a strictly increasing function with $f(0) = 0$. and $x > 0$. The function $A(x) = \\int_0^x f(u) du$ is also plotted. The point $Q$ is $f(x)$, and the point $P$ is $A(x)$. The point $T$ is chosen to so that the length between $T$ and $x$ times the length between $Q$ and $x$ equals the length from $P$ to $x$. ($\\lvert Tx \\rvert \\cdot \\lvert Qx \\rvert = \\lvert Px \\rvert$.) Barrow showed that the line segment $PT$ is tangent to the graph of $A(x)$. This figure illustrates the labeling for some function:\n\n::: {.cell hold='true' execution_count=42}\n\n::: {.cell-output .cell-output-display execution_count=40}\n![](ftc_files/figure-html/cell-43-output-1.svg){}\n:::\n:::\n\n\nThe fact that $\\lvert Tx \\rvert \\cdot \\lvert Qx \\rvert = \\lvert Px \\rvert$ says what in terms of $f(x)$, $A(x)$ and $A'(x)$?\n\n::: {.cell hold='true' execution_count=43}\n\n::: {.cell-output .cell-output-display execution_count=41}\n```{=html}\n
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\n\n\n```\n:::\n:::\n\n\nThe fact that $\\lvert PT \\rvert$ is tangent says what in terms of $f(x)$, $A(x)$ and $A'(x)$?\n\n::: {.cell hold='true' execution_count=44}\n\n::: {.cell-output .cell-output-display execution_count=42}\n```{=html}\n
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\n\n\n```\n:::\n:::\n\n\nSolving, we get:\n\n::: {.cell hold='true' execution_count=45}\n\n::: {.cell-output .cell-output-display execution_count=43}\n```{=html}\n
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\n\n\n```\n:::\n:::\n\n\n###### Question\n\n\nAccording to [Bressoud](http://www.math.harvard.edu/~knill/teaching/math1a_2011/exhibits/bressoud/) \"Newton observes that the rate of change of an accumulated quantity is the rate at which that quantity is accumulating\". Which part of the FTC does this refer to:\n\n::: {.cell hold='true' execution_count=46}\n\n::: {.cell-output .cell-output-display execution_count=44}\n```{=html}\n
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\n\n\n```\n:::\n:::\n\n\n## More on SymPy's `integrate`\n\n\nFinding the value of a definite integral through the fundamental theorem of calculus relies on the algebraic identification of an antiderivative. This is difficult to do by hand and by computer, and is complicated by the fact that not every [elementary ](https://en.wikipedia.org/wiki/Elementary_function)function has an elementary antiderivative. `SymPy`'s documentation on integration indicates that several different means to integrate a function are used internally. As it is of interest here, it is copied with just minor edits below (from an older version of SymPy):\n\n\n#### Simple heuristics (based on pattern matching and integral table):\n\n\n * most frequently used functions (e.g. polynomials, products of trigonometric functions)\n\n\n#### Integration of rational functions:\n\n\n * A complete algorithm for integrating rational functions is implemented (the Lazard-Rioboo-Trager algorithm). The algorithm also uses the partial fraction decomposition algorithm implemented in `apart` as a preprocessor to make this process faster. Note that the integral of a rational function is always elementary, but in general, it may include a `RootSum`.\n\n\n#### Full Risch algorithm:\n\n\n * The Risch algorithm is a complete decision procedure for integrating elementary functions, which means that given any elementary function, it will either compute an elementary antiderivative, or else prove that none exists. Currently, part of transcendental case is implemented, meaning elementary integrals containing exponentials, logarithms, and (soon!) trigonometric functions can be computed. The algebraic case, e.g., functions containing roots, is much more difficult and is not implemented yet.\n * If the routine fails (because the integrand is not elementary, or because a case is not implemented yet), it continues on to the next algorithms below. If the routine proves that the integrals is nonelementary, it still moves on to the algorithms below, because we might be able to find a closed-form solution in terms of special functions. If `risch=true`, however, it will stop here.\n\n\n#### The Meijer G-Function algorithm:\n\n\n * This algorithm works by first rewriting the integrand in terms of very general Meijer G-Function (`meijerg` in `SymPy`), integrating it, and then rewriting the result back, if possible. This algorithm is particularly powerful for definite integrals (which is actually part of a different method of Integral), since it can compute closed-form solutions of definite integrals even when no closed-form indefinite integral exists. But it also is capable of computing many indefinite integrals as well.\n * Another advantage of this method is that it can use some results about the Meijer G-Function to give a result in terms of a Piecewise expression, which allows to express conditionally convergent integrals.\n * Setting `meijerg=true` will cause `integrate` to use only this method.\n\n\n#### The \"manual integration\" algorithm:\n\n\n * This algorithm tries to mimic how a person would find an antiderivative by hand, for example by looking for a substitution or applying integration by parts. This algorithm does not handle as many integrands but can return results in a more familiar form.\n * Sometimes this algorithm can evaluate parts of an integral; in this case `integrate` will try to evaluate the rest of the integrand using the other methods here.\n * Setting `manual=true` will cause `integrate` to use only this method.\n\n\n#### The Heuristic Risch algorithm:\n\n\n * This is a heuristic version of the Risch algorithm, meaning that it is not deterministic. This is tried as a last resort because it can be very slow. It is still used because not enough of the full Risch algorithm is implemented, so that there are still some integrals that can only be computed using this method. The goal is to implement enough of the Risch and Meijer G-function methods so that this can be deleted. Setting `heurisch=true` will cause `integrate` to use only this method. Set `heurisch=false` to not use it.\n\n", "supporting": [ "ftc_files/figure-html" ], "filters": [], "includes": { "include-in-header": [ "\n\n\n" ] } } }