An integral

Source: Alex Bolton (inspired by Book Proofs blog)
What is

Show hint

Show answer & extension

Find them all

Find all continuous positive functions, \(f\) on \([0,1]\) such that:
$$\int_0^1 f(x) dx=1\\ \mathrm{and }\int_0^1 xf(x) dx=\alpha\\ \mathrm{and }\int_0^1 x^2f(x) dx=\alpha^2$$

Show answer & extension


$$\int_0^1 1 dx = 1$$
Find \(a_1\) such that:
$$\int_0^{a_1} x dx = 1$$
Find \(a_2\) such that:
$$\int_0^{a_2} x^2 dx = 1$$
Find \(a_n\) such that (for \(n>0\)):
$$\int_0^{a_n} x^n dx = 1$$

Show answer & extension


Show me a random puzzle
 Most recent collections 

Sunday Afternoon Maths LXVII

Coloured weights
Not Roman numerals

Advent calendar 2018

Sunday Afternoon Maths LXVI

Cryptic crossnumber #2

Sunday Afternoon Maths LXV

Cryptic crossnumber #1
Breaking Chocolate
Square and cube endings

List of all puzzles


ave volume bases logic sequences differentiation balancing prime numbers arrows symmetry remainders integers chocolate addition shape numbers chalkdust crossnumber star numbers dice number probability probabilty colouring money partitions quadratics division functions speed factorials cube numbers wordplay geometry integration pascal's triangle spheres perimeter mean time sums square numbers factors odd numbers algebra squares digits floors menace angles routes irreducible numbers indices games grids trigonometry chess multiples christmas multiplication percentages rugby scales sum to infinity regular shapes fractions people maths graphs circles triangle numbers clocks 3d shapes advent triangles shapes crossnumbers coins dodecagons ellipses books surds polygons complex numbers cryptic clues area means cards hexagons square roots perfect numbers proportion 2d shapes coordinates calculus folding tube maps doubling rectangles sport palindromes parabolas taxicab geometry unit fractions dates averages crosswords lines planes cryptic crossnumbers


Show me a random puzzle
▼ show ▼
© Matthew Scroggs 2019