Integral arranging
How would you sort out these integrals?
Problem
How might you sort these integrals into an order or different groups?
$$ \int\frac{1}{1+x^2}\,dx\quad\quad\quad\int\frac{1}{1-x^2}\,dx\quad\quad\quad\int\frac{1}{(1+x)^2}\,dx\quad\quad\int\frac{1}{(1-x)^2}\,dx $$ $$ \int\frac{1}{1+x}\,dx\quad\quad\quad\int\frac{1}{1-x}\,dx\quad\quad\quad\int\frac{1}{\sqrt{1+x^2}}\,dx\quad\quad\int\frac{1}{\sqrt{1-x}}\,dx $$ $$ \int{\sqrt{1+x^2}}\,dx\quad\quad\int{\sqrt{1-x^2}}\,dx\quad\quad\int \sqrt{1+x}\,dx\quad\quad\int \sqrt{1-x}\,dx $$
Although you can compute many integrals using Wolfram's integrator, if you do enough mathematics you will realise that the class of functions which integrate to a closed algebraic form is, by most ways of counting, small. There are many advanced analytical tools which allow for the manipulation and approximate computation of integrals more generally. A large part of this procedure involves classifying integrals into different types before suitable approximations are made.
Getting Started
There is no particular 'answer' to this question, but here are some points to consider:
- Are there some that you know the answer to, others that you don't?
- Are some equivalent under a change of variables?
- Are some valid over different ranges of integration?
- Are some larger than others over different ranges?
- Are some more complex than others to evaluate?
- Do some seem to have the same 'class' of answer?
You might start by considering those which can be computed using standard rules for integrating powers of $x$ and which cannot. What other options are avilable to you?
Student Solutions
There is no particular 'answer' to this question, but here are some points to consider:
- Are there some that you know the answer to, others that you don't?
- Are some equivalent under a change of variables?
- Are some valid over different ranges of integration?
- Are some larger than others over different ranges?
- Are some more complex than others to evaluate?
- Do some seem to have the same 'class' of answer?
Teachers' Resources
Why do this problem?
This is an opportunity for students to distinguish between very similar looking integrals which might require very different approaches. By grouping the integrals in different ways, students will be more sensitive to the subtle changes linked to different techniques, different values and different ranges of validity.
Possible approach
Students can dive straight into grouping the integrals with little preamble. It is best if this is done in small groups to encourage discussion and diverse approaches. You might start students off by asking them to notice what is the same and what is different about the integrals. Perhaps you could consider how you might group them if you knew nothing about integration (considering factors such as squares, square roots, fractions and addition/subtraction). Further questions for consideration can be found in the Getting Started section and below. To make sorting easier (and more flexible), the integrals could be printed on cards, written on individual whiteboards or on post-it notes.
Key questions
- Are there some that you know the answer to, others that you don't?
- Are some equivalent under a change of variables?
- Are some valid over different ranges of integration?
- Are some larger than others over different ranges?
- Are some more complex than others to evaluate?
- Do some seem to have the same 'class' of answer?
Possible support
Students could use a formula book to separate out "standard" integrals and to help distinguish between inverse trigonometric results and inverse hyperbolic.
Possible extension
Students completing a further mathematics course should be able to compute all of these integrals (with a formula book). The natural extension is to compute them, demonstrating the most efficient approach and noting any limits on their validity.