Integral arranging
How would you sort out these integrals?
Age
16 to 18
Challenge level
Problem
How might you sort these integrals into an order or different groups?
$$ \int\frac{1}{1+x^2}\rm dx\quad\quad\int\frac{1}{1-x^2}\,dx $$ $$ \int\frac{1}{(1+x)^2}\,dx\quad\quad\int\frac{1}{(1-x)^2}\,dx $$ $$ \int\frac{1}{1+x}\,dx\quad\quad\int\frac{1}{1-x}\,dx $$ $$ \int\frac{1}{\sqrt{1+x^2}}\,dx\quad\quad\int\frac{1}{\sqrt{1-x}}\,dx $$ $$ \int{\sqrt{1+x^2}}\,dx\quad\quad\int{\sqrt{1-x^2}}\,dx $$ $$ \int \sqrt{1+x}dx\quad\quad\int \sqrt{1-x}\,dx $$
Did you know ... ?
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.
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.
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?