General integration rules:👇 1. ∫kf(x) dx = k∫f(x) dx 2. ∫[f(x) ± g(x)] dx = ∫f(x) dx ± ∫g(x) dx 3. ∫f(x) g′(x) dx = f(x) g(x) – ∫f′(x) g(x) dx 4. ∫f(g(x)) g′(x) dx = F(g(x)) + C, where F′(u) = f(u) and u = g(x)

William Jones first used π in 1706 in Synopsis Palmariorum Matheseos to denote the ratio of a circle’s circumference to its diameter. The rest is history & wonders around it!

Pursue mathematics as an art form.

Calculate the sum (S) of the series.

What are the values of a and b in the equation?

Sum of Squares.

Euler’s Totient Function, φ(n), counts the number of integers from 1 to n that are coprime with n. Coprime means they share no common factors with n other than 1. Note: Here n is a positive integer, and p₁, p₂, p₃…, pₖ are the distinct prime factors of n.

For those who think solving integrals is not a gentleman's game.

Golden ratio φ is the only positive number which is one more than its reciprocal.

The Fibonacci pattern is a sequence of numbers where each number equals the sum of the two previous numbers. It's denoted as “Fₙ”, is established through a recursive formula, initialised with the starting values F₀=0 and F₁=1, expressed as Fₙ = Fₙ₋₁ + Fₙ₋₂ for n ≥ 2.

Cassini's Identity is a property of Fibonacci numbers, which states that for any integer n, the following holds: Fₙ₊₁Fₙ₋₁ - Fₙ² = (-1)ⁿ where Fₙ is the n-th Fibonacci number, defined as F₀ = 0 ,F₁ = 1, and Fₙ = Fₙ₋₁ + Fₙ₋₂

According to the Prime Number Theorem, the number of prime numbers less than or equal to x, denoted π(x), is given by the approximation: The Graphical representation of this prime counting function is this:

Schematic formula for the expansion of a binomial expression (x+y)ⁿ, where n is a non-negative integer.

The Euler's number 'e': • The number e is the limit value of the expression (1+ 1/n) raised to the nth power, as n increases indefinitely. • If L(a) denotes the area under the curve y = 1/x above the interval [1, a], then e is the location on the x-axis for which L(e) = 1.

Euler is not only the most prolific contributor to the development of mathematics, but he has also given us quite a few symbols that are still commonly used today. These include the following: f(x): Function notation, e: Base of natural logarithms, s: Semiperimeter of a

Common Derivative Formulas for Calculus: 1.d/dx c = 0 2. d/dx xⁿ = nxⁿ⁻¹ 3. d/dx eˣ = eˣ 4. d/dx ln |x| = 1/x, x≠0 5. d/dx sin x = cosx 6. d/dx cos x = –sin x 7. d/dx tan x = sec² x 8. d/dx cot x = –csc² x 9. d/dx sec x = sec x tan x 10. d/dx csc x = –csc x cot x 11.

Useful vector identities and results: