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Age 16 to 18 Challenge Level:

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.