Green Cheat Sheets - Ty Harness
These are the Trent Polytechnic Nottingham green cheat sheets dated 1975 [Alas I don't know the original author(s).] Just the standard intergrals and differentials that you're expected to memorise (but sometimes you were allowed them in an exam.)
Algebra
Trigonometry
Intergration
Differentiation
Laplace Transforms

Intergration Formulae (a and b must be constants)

 An arbitrary constant should be added to each indefinate intergral. Indefinate intergrals can differ by constant. It is assumed that arguments of the logarithms are positive. (34) $$\int \sqrt{a^2 - v^2} dv =$$ (1) $$\int v^n dv = \frac{v^{n+1}}{{n+1}}$$ where $$(n\neq-1)$$ (2) $$\int \frac{1}{{v}} dv = \int v^{-1} dv = ln(v)$$ (35) $$\int \sqrt{v^2 - a^2} dv =$$ (3) $$\int \frac{1}{{av + b}} dv = \frac{1}{a} ln(av+b)$$ where $$(a\neq 0)$$ (4) $$\int e^v dv = e^v$$ , and $$\int e^{av+b} dv = \frac{1}{a} e^{av + b}$$ (36) $$\int \sqrt{a^2 + v^2} dv =$$ (5)$$\int sin(av + b)^n dv = \frac{(av + b)^{n+1}}{a(n+1)}$$ (6)$$\int sin v dv = -cos v$$ (37) $$\int e^{f(v)f'(v) dv = e^{f(v)} }$$ (38) $$\int [f(v)]^n f'(v) dv = \frac{[f(v)]^{n+1}}{n+1}$$ (7)$$\int sin(av + b) dv = -\frac{1}{a} cos(av + b)$$ where $$(a\neq 0)$$ (8)$$\int cos v dv = sin v$$ (39) $$\int \frac{f'(v)}{f(v)} dv = ln[f(v)]$$ (40) $$\int ln(v) dv = v[ln(v)-1]$$ (9)$$\int cos(av + b) dv = \frac{1}{a} sin(av + b)$$ where $$(a\neq 0)$$ (10)$$\int tan v dv = -ln(cosv) = ln(secv)$$ (41) $$\int u \frac{dv}{dx} dx = uv - \int v\frac{du}{dx}dx$$ (42) $$\int e^{av}sin(bv) dv = \frac{e^{av}}{a^2 + b^2}[asin(bv) - bcos(bv)]$$ (11)$$\int cot v dv = ln(sinv)$$ (12)$$\int$$ or $$\int udv = uv - \int vdu$$ (43) $$\int e^{av}cos(bv) dv = \frac{e^{av}}{a^2 + b^2}[acos(bv) - bsin(bv)]$$ (13)$$\int sec v dv = ln(sec v + tan v)$$ (15)$$\int$$ (14)$$\int cosec v dv = - ln$$ (17)$$\int$$ (44) $$\int f(x) dx = \int f(g(u))\frac{dx}{du} du$$ where (45) $$\int_0^\infty e^{v^2} dv = \frac{\sqrt{\pi}}{2}$$ (16)$$\int sin^2v = \frac{1}{2}(v - sin v cos v)$$ (46) $$\int_{0}^{\frac{\pi}{2}} sin^nv dv = \frac{n-1}{n} \int_{0}^{\frac{\pi}{2}} sin^{n-2}vdv$$ for any n (18)$$\int cos^2v = \frac{1}{2}(v + sin v cos v)$$ (48) $$\int_{0}^{\frac{\pi}{2}} sin^mvcos^nv dv = I_{m,n} = \frac{m-1}{m+n} I_{m-2,n} = \frac{n-1}{m+n}I_{m,n-2}$$ (19)$$\int cosecv cotv dv = -cosecv$$ (20)$$\int sinhv dv = coshv$$ (21)$$\int coshv dv = sinh v$$ (22)$$\int sech^2 v dv = tanhv$$ (23)$$\int cosech^2v dv = ln(sinhv)$$ (24)$$\int tanh v = ln(coshv)$$ Numerical Intergration (a) Trapezium rule. One strip $$\int_{x_0}^{x_1} y dx = \frac{h}{2}[y_0 + y_1] + \epsilon$$ (25)$$\int cothv dv = ln(shinhv)$$ (26)$$\int sechv dv = 2tan^{-1}(e^v)$$ (27)$$\int cosechv dv = ln (\frac{e^v-1}{e^v+1})$$ (28)$$\int sechv tanhv dv = -sechv$$ (b) Simpson's rule. Two strip $$\int_x_0$$ (29)$$\int \frac{1}{a^2 + v^2} dv = \frac{1}{a}tan^{-1}(\frac{v}{a})$$ (30)$$\int \frac{1}{\sqrt({a^2 - v^2})} dv = sin^{-1}\frac{v}{a}$$ (31)$$\int \frac{1}{a^2 - v^2} dv = \frac{1}{a}tanh^{-1} (\frac {v}{a})$$ (c) Milne 4 strip $$\int_x_0 ydx = \frac{4h}{3} [2y_1 - y_2 - 2y_3] + \epsilon$$ where (32)$$\int \frac{1}{\sqrt{v^2 + a^2}} dv = sinh^{-1} (\frac{v}{a})$$ or $$ln[v + \sqrt{v^2 + a^2}]$$ (33)$$\int \frac{1}{\sqrt{a^2 - v^2}} dv = cosh^{-1} (\frac{v}{a})$$ or $$ln[v + \sqrt{v^2 - a^2}]$$