Integration Bee | Department of Mathematics

Integration Bee

FUN & PRIZES

Every April, as part of Mathematics Awareness Month, the UNT Department of Mathematics sponsors an INTEGRATION BEE, a contest for undergraduates and high school students with prizes for the best skills in evaluating indefinite integrals. Come spend a fun few hours with other integration enthusiasts. Cake and drinks will be available for participants.

Non-UNT students welcome! Please bring a school photo-id to verify undergraduate (or high school) status.

Rules

The contest consists of elimination rounds in which contestants find antiderivatives of given functions. As the contest progresses, functions increase in complexity.

  • Contestants who wrongly evaluate two integrals are disqualified from succeeding rounds.
  • Results are displayed during the contest only by (confidential) participant numbers; only the winners are made public.
  • The three most skillful contestants will be awarded prizes.
  • All undergraduate and high students are eligible to participate.

2017 Contest Information

  • Date: Friday, April 7
  • Start Time: 3:00 p.m.
  • Room: SAGE 116
  • Registration: To register to participate in the Bee, please e-mail Rita Sears <rhsears@unt.edu>, stop by GAB 443, or call (940) 565-4045.
  • Questions? If you have questions, you can contact Professor Joe Iaia by e-mail at <iaia@unt.edu> or call him at (940) 565-4704
  • Prizes: The 2017 prizes will be \$75, \$50, and \$25 tuition credits that can be used toward UNT tuition.

Integration Techniques Everyone Should Know

  • Basic techniques:
    • Integrals of powers and of trigonometric functions.
    • Substitution.
    • Integration by parts.
    • Partial fraction decomposition.
    • Trigonometric substitutions.
  • Tricky stuff:
    • Integrals of powers of trigonometric functions.
    • Integrals of hyperbolic functions.
    • Completing the square.
    • Elimination of radicals by substitution.
    • Substituting the tangent of a half-angle.

Example Problems

Example Problems
To see a hint on how to do an integral, move your mouse cursor over the integral
$\displaystyle\int\frac{x^2+2x+1}{x^4}\,dx$
$\displaystyle\int\frac{\sin\sqrt{x}}{\sqrt{x}}\,dx$
$\displaystyle\int x^2\sqrt{x+4}\,dx$
$\displaystyle\int{x}^3e^{-x^2}\,dx$
$\displaystyle\int\frac{dx}{x^2+x+1}\,dx$
$\displaystyle\int\sqrt{1-e^x}\,dx$
$\displaystyle\int\frac{x}{2+e^x+e^{-x}}\,dx$
$\displaystyle\int\sec^5x\,dx$
$\displaystyle\int\frac{dx}{\sqrt{1-x^2+\arcsin x-x^2\arcsin x}}$
$\displaystyle\int\frac{1+\sin x}{1+\cos x}\,dx$
$\displaystyle\int\frac{dx}{(x^2+4)^3}$
$\displaystyle\int\frac{dx}{x^4-16}$
$\displaystyle\int\frac{\cos\ln{x}}{x}\,dx$
$\displaystyle\int\frac{(x-1)^3}{x^2}\,dx$
$\displaystyle\int\frac{x^2\,dx}{\sqrt{x-1}}$
$\displaystyle\int{x}^3\sin(x^2)\,dx$
$\displaystyle\int\sqrt{x}\ln{x}\,dx$
$\displaystyle\int\frac{dx}{\sin{x}+\cos{x}}$
$\displaystyle\int{e^x\sin{x}\,dx}$
$\displaystyle\int\frac{x^2\,dx}{(x^2+8)^{3/2}}$
$\displaystyle\int\frac{dx}{x^3+8}$
$\displaystyle\int\frac{dx}{x\ln{x}}$
$\displaystyle\int\frac{x^2\,dx}{\sqrt{x+3}}$
$\displaystyle\int{x^3e^{x^2}}\,dx$

The awful truth about antiderivatives

All of the integrands in the Integration Bee contest are elementary functions, that is, finite combinations of algebraic, trigonometric, and exponential functions, and their inverses. So are all of their antiderivatives.

But the awful truth is that many (in some sense most) elementary functions have antiderivatives that cannot be expressed in terms of elementary functions. For example, although you can find an explicit formula for an antiderivative of $\sqrt{1+x^2}$, there is no explicit formula in terms of elementary functions for an antiderivative of $\sqrt{1+x^4}$.

For information about why there is no such formula in general, see:

  • E Marchisotto & G Zakeri, "An invitation to integration in finite terms", COLLEGE MATHEMATICS JOURNAL (1994), pp 295-308.
  • Matthew P Wiener, Functions without elementary antiderivative, sci.math usenet posting 1997/11/30.

There are many simple-looking functions that have no elementary-function antiderivative. Such nonelementary integrals include:$$\begin{array}{lll}\displaystyle\int\frac{e^x}{x}\,dx,\qquad&\displaystyle\int\frac{\sin x}{x}\,dx,\qquad&\displaystyle{\int}e^{x^2}\,dx,\\ \displaystyle\int\frac{1}{\ln x}\,dx,\qquad&\displaystyle\int\ln(\ln x)\,dx,\qquad&\displaystyle\int\sin(x^2)\,dx\\ \displaystyle{\int}e^{e^x}\,dx,\qquad&\displaystyle\int\sqrt{1+x^4}\,dx,\qquad&\text{and}\quad\int\sqrt{2-\sin^2x}\,dx\end{array}$$

No matter how hard you may try, you will not discover explicit elementary-function formulas for these indefinite integrals.

Since there is only a small lexicographical difference between $\displaystyle\int\sin(x^2)\,dx$, which cannot be expressed in terms of elementary functions, and $\displaystyle{\int}x\sin(x^2)\,dx$, which can easily be evaluated, you can see that antidifferentiation is a touchy business. (All the Integration Bee problems have been carefully rigged so that the art of finding explicit antiderivatives can be successfully applied.)

The skill celebrated by the Integration Bee contest is thus not broadly applicable to indefinite integrals. In general, all we know about the integral of an elementary function is the assertion of the Fundamental Theorem of Calculus: Every continuous function has an antiderivative. (But there is no guarantee we can find a formula for an antiderivative in terms of elementary functions like sine, cosine, logarithm, and so forth.)

Recent Prize Winners

2017

Zach Thompson (TAMS), Pranav Murugan (TAMS), and Austin McGregor

2016

Benjamin Zheng (TAMS), Audrey Parker (TWU), and Tony Gao (TAMS)
2015
Tony Liu (TAMS), Anirban Sarkar (TAMS) and Bryan Xie (TAMS)
2014
Mohammad Behnia (TAMS), Chia Ting Han (UNT Senior) and Chris Shen (TAMS)
2013
Chris James (UNT Senior), Stacy Ho (TAMS) and Kevin Yang (TAMS)
2012
Trevor Davila, Chenyao Yu and Enkhjargal Lkhagvajov

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