Balloon Calculus!



Standard integrals, derivatives and methods



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 tan x
differentiate tan x
Usually, you'll integrate sec2 x by remembering that it's the derivative of tan x. If you want to forego that short-cut and integrate strategically, you can use the Weierstrass sub - see for example the integrate cosec section 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

Gallery









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

Gallery










Differentiate a quotient
differentiate a quotient


Note: the function 1/g would be written g/g2 in the bottom row, just so we have a common denominator ready for simplifying the row... very like the standard quotient-rule formula. But, note also that if g is a power of a function (e.g. if it's sin2x) then we'll put sinx in the dashed balloon and write the second balloon in the bottom row as sinx over sin3x. This is (in these cases) a bit neater than the formula, which produces sin4x in the denominator.

Examples of differentiating a quotient (2x4 - 3x) / (4x - 1) 1/6 e2 sin x - 3x / (3 + 2 cos x) (1 - x2)3 / (6 + 2x)3 (2x - 3)2 / (x3 - 7)3 x (x2)/(x + 2) (1 - x2)/(x2 + 1)2 (x + 5)/(2x - 4) -4x/(x2 - 1)2 (1 + x2)/(1 + 2x2) 5 cosx/(4 - secx) x/(x + 2)2 x2/e-2x tan t (exponential form) ln|x| / √x √(7x - 2) / √(5x + 3) (4x + 3)7 / (5x - 2)7 3x/(x2 + 4) (4 - x2) / [ (x2 + 4) ]2 (1-s2 ) / (1 + s2 )2

Gallery










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)

Gallery


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

differentiate an inverse function









Integration by parts
integration by parts
Examples of integration by parts ∫ 5x sin(2x) dx ∫ secx(2x + x2 tanx) dx x2 e4x dx ex sinhx dx x / √(2x + 1) dx x9 √(x5 - 3) dx x3 √(2 + x2) dx ∫ 2x3 e-x2 dx ∫ (1 - 2 cosx) (3 + 2 cosx) e(1/2 x + cosx dx x2 ln|4x| dx ∫ arctanx dx ∫ (5x4 + 4x5) / (x5 + x + 1) dx ∫ (x + 2) sin(x) dx          ∫ cos(2x)(5x + 1) dx u dv = uv - ∫ v du ∫ √x ln(x) dx x3 ex2 dx ∫ 1 / [(x - 1)3/4 (x + 2)5/4] dx ∫ arcsinx / √(1 + x) dx ∫ 1 / [x2 (x4 + 1)3/4] dx ∫ 2x √(2x - 3) dx ∫ ln(2x)/x2 dx ∫ 1/x e1/x dx ∫ cos2x dx x / (2x + 1)4 dx ∫ sin^2 x dx ∫ ln(lnx) dx ∫ (lnx)2/x2 dx ∫ √ (1 - x2) dx x arctanx dx x3 √ (x2 + 1) dx ∫ sin2x / cos5x dx ∫ (1/3)u sinu du ∫ (1/5)u cos(u) du ∫ (2 sin x - 1) (sin x + 2) e2/3 sin x dx ∫ √(a2 - x2) dx ∫ (x + 2) (1 - e-x/3) dx ∫ 5x e4x dx ∫ ln (x2 - x + 2) dx ∫ ln (x + 1) dx ∫ ln (2x + 1) dx

Parts twice ∫ √(x - √(x2 - 4)) dx e(2x) sinx dx ex cosx dx ∫ 3x2 ex/2 dx ∫ cos(lnx) dx x2 cos (π x) dx ex sin(2x) dx x2 sin(3x + 1) dx

Parts thrice
∫ (2x3 + 5x + 1) e2x dx x3 sin x dx

Parts using triple product rule 01 lnx ln(1 - x) dx

Recurrence formulae yn = ∫ 01 tne-t dt, n = 1, 2, 3,...

Gallery










Integration by substitution / change of variable
integration by substitution
Examples of integration by substitution ∫ (ex - 1) / e2x dx ∫ 1 / (64x2 + 81) dx x / (1 + x2) dx ∫ (v - 1) / (v2 + 2v + 2) dv x3 √(2 + x2) dx ∫ sin2(4t) cos(t) dt u dx ∫ 1/3x3 √(1 + x4) dx x / √(4 - x2) dx ∫ sinx / cos2x dx e- x dx ∫ 1 / (tan 2x [1 - 2 ln(sin 2x)]) dx ∫ (ex + 1) / (ex - 1) dx x2 √ (7 - 3x3) dx ∫ sec2(3x) dx ∫ √ x / ( 1 + x1/4) dx ∫ arcsin √x / √ (x - x2) dx ∫ (5x2 + 4)3 dx ∫ (lnx)2/x2 dx ∫ 1 / (e2x + 1) dx z3 e-z2/2 dz x ex2 dx 0a2/25 x24 √(a2 - x25) dx 01 (ez + 21) / (ez + 21z) dz ∫ 1 / (9 sin2x + 4 cos2x) dx ∫ 1 / [x (1 - 4x)] dx ∫ 2x/(3 + x4) dx esin x sin(2x) dx x2 √(2 + x3) dx p(p + 1)5 dp t/(t2 + 2) dt t3 √(t2 + 1) dt ∫ √x / (1 + x) dx ∫ 1 / (√x (1 + x)) dx ∫ 8(y - 9)-3 dy ∫ 3x e2x2 - 1 dx e- x dx ∫ sec2x / (4 + tan2x) dx ∫ sec x tan x dx ∫ 1 / [(a - x)(b - x)] dx ∫ ln | 1 - x | dx ∫ cos(a - x) dx ∫ tan5x sec4x dx ∫ cos4(3t) sin(3t) dt 0π / 3 sin θ / cos2 θ dθ ∫ 4t / (1 + 16t2) dt ∫ - 4x / (2x2 + 1)2 dx ∫ 6 sec2(5x) / [7 - 3 tan(5x)]7 dx ∫ ½ ex / √x dx 13 (x + 3) / (x2 + 6x) dx 0π sin t / (1 + cos2 t) dt y √(1 - y) dy

Gallery










Integration by internal substitution
integration by internal substitution
Examples of integration by internal substitution ∫ 1 / (1 + x2) dx ∫ 1 / [(x2 + a2) √(x2 + a2 + b2) dx ∫ 1/(5 - 2√x) dx ∫ √(a2 + x2) dx 12 1/√(4 - x2) dx ∫ (4 - x) / √(4 - x2) dx ∫ 6 / (9x2 + 4) dx x3 √ (x - 3) dx ∫ √ x / ( 1 + x1/4) dx ∫ 1/(x2 + x + 1) dx ∫ 1/(3 + x2) dx ∫ √ (6 - 6x2) dx 01 1 / (1 + 3x) dx ∫ 1 / √ (x2 + 25) dx ∫ 1/2 x2 √ (1 + x2) dx ∫ √ (1 + 1/x2) dx          ∫ √ (1 + 4x2) dx          ∫ √ (1 - (x/a)2) dx ∫ 1 / (c2 + x2)(3/2) dy ∫ √(4 - x2) dx ∫ 1 / √(49 + x2) dx ∫ 1 / (x2 + 1)2 dx ∫ 1 / (x2 √(x2 - a2) dx ∫ 1 / √(1 - x2) dx

Weierstrass substitution ∫ 2 / (2 cosx + sinx) dx ∫ secx dx ∫ cosecx dx

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Separation of variables

Get your equation in the form f(y) * dy/dx = f(x). (Not a complete separation). Fit the equation along the bottom row...

separation of variables

You can then extend the diagram on either side if necessary (chain-rule shapes shown for example).

separation of variables
Examples of separation of variables rate of evaporation from a dish of water x dy/dx - 3y + 2 = 0 dy/dx = -y / (4 + x) dP/dt = kP dF/dt = -2F 1/P dP/dt = k dy/dx = 2y (100 - t^2) dN/dt + 4t N = 0 dh/dt = (6 - h) /20 dy/dx = Cy

Gallery










Integrating factor

If you can get your equation in the form      dy/dx + y f(x) = 0      or      dy/dx + y f(x) = g(x),

then fill just these places in the bottom row...

integrating factor

... and the others will follow...
integrating factor

Note: in place of y you can accomodate a function f(y)... as long as, in the place of dy/dx, you have d (f(y)) / dx.

Examples of choosing an integrating factor (x3 + y) dx - x dy = 0 birth and death rates ex dy/dx = 1 - 2exy dy/dx = 2x (1 + x2 - y) dy/dx = y/x + y2/x2 x dy/dx - 4y = 0 dθ/dt + θ/10 = 5 - 5t/2 x2 y' + xy = x ex dy/dx - y / (x + 1) = x - 1

Gallery










Exact equations

If you can get the left-side of your equation to fit the lower level of the product rule without need of an integrating factor (see above), the equation is exact. Most examples of 'exact equations', however, don't necessarily fit the product rule as such, but do fit the multi-variable chain rule...

exact equation

... of which the product rule is a special case. To be able to find F you need an equation that fits the bottom row and also passes the 'exactness test'. That is, you need to check that the partial derivative of M with respect to y equals that of N with respect to x. That settled, you know you can integrate as follows...


More examples of solving an exact equation 3x2/y2 + y2/x2 - 2/x3 6/y4 - 2/x3 + (6/y4 - 2x3/y3 - 2y/x) y' = 0 u/∂x . ∂u/∂y = xy F(x,y,z) = (yz2 - 2)i + (xz2 - e2x - 6y2)j + (2xyz + 4z - 2y e2x)k











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

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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








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