Area L

Draw the graph of a continuous increasing function in the first quadrant and horizontal and vertical lines through two points. The areas in your sketch lead to a useful formula for finding integrals.

Integral Equation

Solve this integral equation.

Integral Sandwich

Generalise this inequality involving integrals.

Weekly Challenge 45: Integral Arranging

Stage: 5 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.