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.

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.