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 

Advent calendar 2019

Sunday Afternoon Maths LXVII

Coloured weights
Not Roman numerals

Advent calendar 2018

Sunday Afternoon Maths LXVI

Cryptic crossnumber #2

List of all puzzles


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


Show me a random puzzle
▼ show ▼
© Matthew Scroggs 2012–2020