Balloon Calculus!


Standard integrals, derivatives and methods



Garden City Maths - Online tuition! Find me on Zoom - just open a meeting and invite through email. Or just email, and I'll help set up.

email: tom@ballooncalculus.org.

Newtonian Mechanics - interactive!
Integrand/derivative
Integral/differand


sec2 x tan x


-cosec x cot x

sec x tan x

- cosec2 x
cosec x

sec x

cot x


1/sqrt(1-x^2) arctan x
1/sqrt(1-x^2) arcsin x
1/sqrt(x^2-1) arccos x


tan x ln | sec x |


cosec x ln | tan (x/2) |
sec x ln | sec x + tan x |
cot x ln | sin x |




Methods of Integration and Differentiation
Differentiate a product
Differentiate a composite function (chain rule)
Differentiate a quotient
Differentiate an inverse trig function
Integration by parts
Integration by substitution / change of variable
Integration by internal substitution
Separation of variables
Integrating factor
Exact equations
Reduction of Order
Fundamental theorem of calculus
Double and Triple Integrals









Given these pairs (either differentiating down or, adding a constant of integration, integrating up)...
x^n, e^x etc.
... along with the chain rule...
x^n, e^x etc.
... and the product rule...

x^n, e^x etc.

... we illustrate some common results below...
















Derivative of cosec x
Derivative of cosec x






Derivative of sec x
Derivative of sec x






Derivative of cot x
Derivative of cot x










Derivative of arctan x
derivative of arctan x






Derivative of arcsin x
Derivative of arcsin x






Derivative of arccos x
Derivative of arccos x










Integral of tan x
Integral of tan x

Alternatively...

Integral of tan x










Integral of cosec x
Integral of cosec x


Alternatively, using the Weierstrass substitution, which is more round about but also a more widely applicable method...


Weierstrass integral of cosec x




Integral of sec x
Integral of sec x


Alternatively, using the Weierstrass substitution, which is more round about but also a more widely applicable method...




Integral of cot x
Integral of cot x








Methods of Integration and Differentiation




Differentiate a product

product rule

... is the product rule. The straight lines differentiate downwards (integrate up) with respect to x (or whatever the main, explicit variable). And then, because of the product rule, the whole of the bottom row is the derivative (with respect to x) of the whole of the top row. Choosing legs crossed or uncrossed is unimportant when differentiating but crucial for integration by parts (see below).



So differentiation is a downwards journey, and if you want to expand the picture with equals signs (often a good idea) then it's probably also clockwise...
differentiate a product

Examples of differentiating a product (3x2 - 3) (x2 - 2) / (x2 + 2) x e-kx relativisitic momentum x arctany d2/dt2 et cos t m ln m + em tan m (1 - e-x2) (1 - e-y2) 2x/(y2 + 2) ∂/∂x, ∂/∂y... ex siny f(x - y) x2 e-kx (5x3 + 1)8 (4x5 + 3)7 -8x-x2 tanx ln|2x + 5| t (1 - t2) x2 e-3e x2 (x - 2)2 3x e2x cos4(3t) sin(3t) 2x  ln |x2 + 5| x(1 - x)(3/5)

Related rates production level and price total & average income price, sales and revenue







Differentiate a composite function

chain rule
... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (which is the inner function of the composite expression which is subject to the chain rule).

As with the product rule, differentiation is generally a clockwise journey...
differentiate a composite

Examples of differentiating a composite function ln(tanx) differentiate integral of composite directional derivatives sin3(cos2(3ex2 )) ∂/∂x √ (x2 + y2) f(4x3) 1 / (1 + 12.6e-0.73x) Lorentz factor 3(2 - 8x)4 wave equation 3ex2 sin(e3x) sin2(sin(sinx)) tan2(x3) 5x3 - 3/(2 - 7x) + 3x5 - 15 sec3(π /2 - x) ekx δ / δx exy √(x + √(x + √(x))) arcsin3(2x) [ sin( 1 + {cos( 1 + [tan(1+x)]4) }3) ]2 [(4x + 3) / (5x - 2)]7 esec x (6x4 - 6x -3 - 2x + 5) -2 - ln | cos x | ln | 8x3 - 7x + 2.78 |

Related rates snowball melting resisters in parallel shadow against a wall revenue given rate for cost ladder sliding down wall rate y 12 times rate x tracking balloon rise shadow length at noon camera tracking ferris wheel highway patrol plane area of inkstain track climbing aircraft aeroplanes converging piston, crank, connecting rod outer rate 12 times inner camera tracking train flow through conical tank rope lifting pipe from far end rotating light-house beam coffee filling in truncated cone rotating spot-light ladder towards wall inflating balloon conical sand pile

Double composite
partial derivative proof d/dx 4/3 x3/4 - x Φ(u,v) = f(uv cos v, uv sin v z(u,v) = f(u2 - v2, 2uv) x = u(s,t), y = v(s,t) d/dx (x2)ex d/dt (5 - 3t)4t d/dx (tan x)x d/dx x (1/x) d/dx f(x) g(x) d/dx x sinx d/dx 10x ln x ∂/∂x (x2 - y2) / (4xy3)

Logarithmic differentiation d/dx cosx lnx / x2 d/dx 4/3 x3/4 - x y = 5x2x tan(6x) d/dx x7x

Implicit differentiation x2 + wx + w2 = 1 x2y3 + 2y = 6 x2y + 3x - 2y = 6 y + ey = x - e2x2 y + x ln y = cos(xy2) 2x/(y2 + 2) y + 2x siny = ex d/dt √(x2 + y2) x2y + xy2 = 6 4x3 + 2xy - y3 = 1/2 x ey = y - 1 ex2 + y6 = 2x cos y x2 + y2 = 25 x3y / (1 - xy) 2x2y = sin(2x) x2y = x + y2 sin3x cos2y = 6 3x2 + 4xy3 = 9 6x + 10y dy/dx - 6 + 20 dy/dx x2 - xy + ¾ x2 = 7

Internal substitution differentiate an inverse














Differentiate an inverse trig function
differentiate an inverse trig function

Examples of differentiating an inverse trig function d/dx arctan(y/x) d/dx arctan(4 - 5x) d/dt arcsec(2√t)

Similarly, to justify the general formula for differentiating an inverse: d/dx f  -1(x) = 1 / [f '(f  -1(x))]...

differentiate an inverse function













































Reduction of Order


Reduction of Order


Examples of reduction of order x2 d2y/dx2 - (2x + 2x2) dy/dx + (2 + 2x) y = 0











Fundamental theorem of calculus


Fundamental theorem of calculus


Examples of fundamental theorem problems d/dxcosxsinx t2 + 2t dt d/dx2x 3 + x3 sin(t2) dt d/dx0ln x √(1 + et) dt d/dx & d/dt: ∫ 0x/√t e-s2 ds d/dxsinxcosx (3 + v2)10 dv d/dx 1 / √(1 + x4) ∫ 1x √(1 + x4) dt d/dx1x2 t4 + t-2 + 1 dt d/dx3cosx et2 dt d/dxx6 cos(√(2t)) dt d/dx0x f(t) dt where I = [f(t)]2 d/dx0ln(x) sin (et) dt more








Double and Triple Integrals


Double and Triple Integrals


Examples of double and triple integration surface of cone inside cylinder truncated cone 0500arcsin(3/5) ½ ρ3 sin(2 φ) dφ dθ dρ 0101 (1 - e-x2) (1 - e-y2) dy dx 0π06 sinθr26r sinθ rz dz dr dθ -π/2π/20a cos θ0√(a2 - r2) r dz dr dθ 0010√(4 - r2) r dz dr dθ 0π/4√ 3√ 3 cos θ r dr dθ 01r2x 2x2y dy dx











Differentiation Under the Integral Sign


Differentiation under the integral sign


Examples of differentiation under the integral sign 01 (x3 - 1) / lnx dx








© Copyright ballooncalculus.org. Email tom@ballooncalculus.org