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


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


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