# Balloon Calculus!

## Standard integrals, derivatives and methods

##### Integral/differand

 sec2 x tan x

 -cosec x cot x sec x tan x - cosec2 x cosec x sec x cot x arctan x arcsin x 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 |

 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)... ... along with the chain rule... ... and the product rule... ... we illustrate some common results below...

##### Differentiate a product ... 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... 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)

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##### Differentiate a composite function ... 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... 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 |

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

##### Integration by substitution / change of variable 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

##### 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