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Welcome to NRICH.

Calculus

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Some of the mathematical symbols used in this index do not display correctly in all browsers. Apologies for any inconvenience.

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This article has some of the ideas used later in calculus when looking at turning points.

• 3 Article: Where do we get our feet wet?

Methods of differentiation and integration

• 4 The Gradient Function: beginner calculus
• 4 Beginner calculus
• 4 Differentiation for coursework on surface area of a box
• 5 Teaching and learning calculus
• 5 Differentiation Explanation
• 5 What does "integrate" mean?
• 5 lim _x->p/4 (tan2x)^tanx
• 5 Maxima and Minima
• 5 Point of Inflexion
• 5 Differentiation for open box problem
• 5 Why do they work: chain rule, product rule, integration by parts?
• 5 Derivative of xⁿ
• 5 Differentiation of xⁿ: proof for real values of n
• 5 Integration by substitution
• 5 Integration by substitution
• 5 Validity of integration by substitution
• 5 Constants of integration
• 5 Integration of algebraic fractions
• 5 Integration of cosec, cosech, etc
• 5 Integrating sec, sech, cosec and cosech (cosh)
• 5 Two different answers to an integral - or not?
• 5 Differentiating implicitly twice
• 5 Volume and Integration
• 5 Volume of revolution
• 5 Surface area of revolution: why does it work?
• 5 Surface area of a sphere by integration
• 5 Reduction Formula (Integration)
• 5 Differentiating cross product
• 5 Length of a curve
• 5 Lengths curves
• 5 Integrating over a region
• 5 Polar coordinates: area by integration
• 5 Integration to calculate Volume
• 6 A continuous function, injective on irrationals injective
• 6 Everywhere continuous, nowhere differentiable function
• 6 A non-integrable derivative
• 6 Proof of the Chain Rule
• 6 Partial differentiation
• 6 Contour Integration
• 6 Infinite Integrals
• 6 Infinite Series as Integrals
• 6 Integral Formula
• 6 L'Hopital's Rule

Particular difficult differentiations and integrations

Some of the mathematical symbols used in this index do not display correctly in all browsers. Apologies for any inconvenience.

• 5 Differientiating |x|
• 5 Derivative of (sinx+cosx)^1/2
• 5 Differentiation of 2^x
• 5 dⁿy/dxⁿ=-ky
• 5 Differentiation of x^x
• 5 ò tan x dx
• 5 òsinx(1 + cos²x) dx
• 5 ò_2.5 ^1.5 (4x² - 9)^1/2 dx
• 5 ò 1/((3+x²)(1+x)^1/2) dx
• 5 ò 2x^1/2.[1 + (1/x)]^1/2 dx
• 5 ò_-(p/4) ^(p/4) sin(x³)dx without a calulator
• 5 ò(2 + 2CosX)^1/2 dx
• 5 ò cot⁴(x) dx
• 5 òsinx.cos3x dx
• 5 ò e^x2 dx
• 5 ò 1/(1+x⁴) dx
• 5 ò x^2-n/(1+x²) dx
• 5 ò cos(x²) dx
• 5 ò x²(x³+2)⁷ dx
• 5 ò(arcSin x)/(1-x²)^½ dx and ò(arcSinh x)/(1+x²)^½ dx
• 5 òcosec(x) dx
• 5 ò sin³2x dx
• 5 Sin²ⁿx
• 5 òSin(x)Cos(x)dx, ò1/(xlnx)dx, òxe^x2dx
• 5 ò₀ ¹x^-xdx
• 5 ò Cosh²x dx and hyperbolic identities
• 5 ò₀ ¹log(1+x)
• 5 y'=òÖ(1+(y')²)
• 5 S^¥ _{n = 1} tan^-1[2/n²] = 3p/4
• 5 ò 1/(x + Ö(1-x²))
• 5 2G(n/2+1)ò₀ ^p/2cosⁿqdq=ÖpG((n+1)/2)
• 5 ò₀ ^p/2 x cos²ⁿ(x) dx
• 6 Integral of Gaussian function (ò e^-x2 dx)
• 6 ò_-¥^¥e^-x2dx
• 6 ò 1/(ln x) dx
• 6 òe^x/x
• 6 ò2x^1/2sin(x) dx
• 6 ò₀ ^¥[1/(1+xⁿ)]dx = p/(nsin(p/n))
• 6 ò₀ ^¥[x³/(e^x-1)]dx
• 5 Hard integrals, and strategies for integration
• 6 ò_-¥ ^¥e^-x2dx=pÖ2
• 6 ò₀^¥ x²/(x⁴+1) dx=p/(2Ö2)
• 6 ò₀ ^2p ln(cos²(x)+1) dx
• 6 Gaussian Integral
• 6 ò₀ ^¥1/(1+xⁿ) dx = p/(n sin p/n)
• 6 Contour integral
• 6 Unintegrable Functions

Differential equations

• 5 Solving a differential equation
• 5 Two differential equations
• 5 [y - x(dy/dx)]/[y²] = cos2x
• 5 1st order Differential Equations: 2 possible particular solutions
• 5 Substitution in a differential equation
• 5 Special solutions to differential equations
• 5 Solving second order differential equations
• 5 Reasons for exponential solutions of differential equations.
• 5 Differential equation (separable variables)
• 5 Differential equations for melting snowman
• 5 dV/dt=-kV^2/3
• 5 d²x/dt²=-w²x
• 5 dv/ds = -(g+kv²)/v
• 5 dy/dx=yx
• 5 dy/dx=(x+y+5)/(2x+3y+13)
• 5 dy/dx=f(y)+g(x)
• 5 dy/dt +ay +b =0
• 5 f'(x)=f(-x)
• 5 First and second degree differential equations
• 5 Two First Order Differential Equations
• 5 Lokta-Volterra equations
• 5 Battles in relation to calculus
• 5 Differentiation minimum area problem
• 5 The Open Box Problem
• 6 Integral equations
• 6 Partial differential equations an Bessel's ODE
• 6 Differential equation in variations problem
• 6 Functional Equation without Real Solutions

Special functions

• 6 Series for Y
• 6 Why does (1/2)! = sqrt(p)/2?
• 6 Gamma and Beta functions
• 6 Lambert's W Function
• 6 The Delta Function
• 6 The Gamma Function
• 6 Approximating the Gamma Function
• 6 P z(x) Convergent

Measure theory

• 6 Measure Theory Discussion
• 6 Fubini's Theorem
• 6 Lebesgue integration
• 6 Half the real numbers
• 6 New Coordinate System
• 6 Measure theory conjecture

Fractional Calculus

This is something the team hadn't heard of until the discussion below!

• 6 Half Derivatives
• 6 Article: Fractional Calculus I
• 6 Article: Fractional Calculus II
• 6 Article: Fractional Calculus III
• 6 Differentiation of fractional differentiation

Miscellaneous

• 5 Bird's Path
• 5 Calculus notation
• 5 Perimeter of curve x^2/3 + y^2/3 = 1
• 5 Length of a curve?
• 5 Proving MacLaurin's theorem
• 5 Maclaurin's Theorem
• 5 Evaluating p
• 5 Big O and little o notation
• 6 Asymptotic and the big O
• 6 "cannot be integrated" - what does it mean?
• 6 Transforms
• 6 What is a Laplace transform?
• 6 Laplace transform for 1/x?
• 6 Laplace Transforms of 3exp(-4t)sin(2t)
• 6 Lagrange Multipliers
• 6 Two continuity questions
• 6 Differentiable Þ Continuous
• 6 Convergence of Fourier series
• 6 Directional Derivaties
• 6 Calculus of variations: a detailed introduction
• 6 The equation of a doughnut (and volume/surface area)
• 6 Ñ . ( f a) = f Ñ .a + a.Ñ f
• 6 Higher Dimensional Mean Value theorem