mscroggs.co.uk
mscroggs.co.uk

subscribe

Puzzles

An integral

Source: Alex Bolton (inspired by Book Proofs blog)
What is
$$\int_0^{\frac\pi2}\frac1{1+\tan^a(x)}\,dx?$$

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

Integrals

$$\int_0^1 1 dx = 1$$
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

Double derivative

What is
$$\frac{d}{dy}\left(\frac{dy}{dx}\right)$$
when:
(i) \(y=x\)
(ii) \(y=x^2\)
(iii) \(y=x^3\)
(iv) \(y=x^n\)
(v) \(y=e^x\)
(vi) \(y=\sin(x)\)?

Show answer & extension

Differentiate this

$$f(x)=e^{x^{ \frac{\ln{\left(\ln{x}\right)}}{ \ln{x}}} }$$
Find \(f'(x)\).

Show answer

x to the power of x again

Let \(y=x^{x^{x^{x^{...}}}}\) [\(x\) to the power of (\(x\) to the power of (\(x\) to the power of (\(x\) to the power of ...))) with an infinite number of \(x\)s]. What is \(\frac{dy}{dx}\)?

Show answer & extension

Archive

Show me a random puzzle
 Most recent collections 

Advent calendar 2024

Advent calendar 2023

Advent calendar 2022

Advent calendar 2021


List of all puzzles

Tags

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

Archive

Show me a random puzzle
▼ show ▼
© Matthew Scroggs 2012–2025