UCC Mathematical Tables

CalculusUCC Mathematical Tablesf(x)x nln |x|cosxsin xtan xsec xcosecxcotxe xe axa xcos −1 x asin −1 x atan −1 x asec −1 x acosec −1 x af ′ (x)f(x) f ′ (x)ae axa x cosh −1 1x √lnax2 − 11− √ tanh −1 1xa2 − x 21 − x 21√ sech −1 1x −a2 − xa2x √ 1 − x 2cosech −1 1x −a 2 + xa2x √ x 2 + 1x √ x 2 − aa2 coth −1 x − 1x 2 − 1−x √ x 2 − a 2nx n−1cot1−aa a 2 + x 2xsinhx coshx− sinx coshx sinhxcosxtanhx sech 2 xsec 2 xsechx − sechxtanh xsecxtan x cosechx − cosechxcoth x− cosecxcotx cothx − cosech 2 x− cosec 2 xe xsinh −1 1x √x2 + 1Product rule y = uv ⇒ dydx = udv dx + vdu dxQuotient rule y = u v ⇒ dy v dudx = dx − udv dxf(x)x n (n ≠ −1)1xcosxsinxtanxsecxcosecx∫f(x)dxx n+1n + 1ln |x|sin x− cosxln | secx|ln | secx + tanx|∣ln∣tan x ∣2cotx ln | sin x|e x e xe ax 1a xa eaxa xlna1√ sin −1 xa2 − x 2 a1 1 xx 2 + a 2 a tan−1 a1x √ 1 xx 2 − a 2 a sec−1 aNewton-Raphson x n+1 = x n − f(x n)f ′ (x n )∫f(x) f(x)dx1√ ln∣ x 2 + a 2∣x2 + a 2 a1 1∣ ∣∣a 2 − x 2 2a ln a + x∣a − x1√ ln∣ x 2 − a 2∣x2 − a 2 asinhx coshxcoshx sinhxtanhx ln coshxcoth x ln | sinhx|sechx tan −1 (sinhx)∣cosechx ln∣tanh ∣2cos 2 1x2 (x + 1 2sin 2x)sin 2 1x2 (x − 1 2sin 2x)cosh 2 1x2 (x + 1 2sinh 2x)sinh 2 1x2 (−x + 1 21sinh2x)− 1 xa sech−1 ax √ a 2 − x 21x √ x 2 + a 2v 2 Taylor series (centre a) f(a + x) = f(a) + f ′ (a)x + f ′′ (a)2!Chain rule f(x) = u ( v(x) ) ⇒ f ′ (x) = du dvMaclaurin series f(x) = f(0) + f ′ (0)x + f ′′ (0)dv dx2!∫ ∫∫Volume of solid ofx=bIntegration by parts u dv = uv − v duV = πy 2 dxrevolution about x-axisx=aTrapezoidal rule A ≈ h []y 1 + y n + 2(y 2 + y 3 + · · · + y n−1 ) Simpson’s rule (n odd) A ≈ h 23− 1 a cosech−1 xax 2 + · · · + f(r) (a)x r + · · ·r!x 2 + · · · + f(r) (0)x r + · · ·r![y 1 + y n + 2(y 3 + y 5 + · · · + y n−2 ) + 4(y 2 + y 4 + · · · + y n−1 )]y 1 y 2 y 3 y 4 y nh

TrigonometrytanA = sin AcosAcotA = cosAsinA = 1tan AsecA = 1cosAcosecA = 1sin Acos(−A) = cosAcos 2 A + sin 2 A = 1 cos2A = cos 2 A − sin 2 A 2 cosAcosB = cos(A + B) + cos(A − B)sec 2 A = 1 + tan 2 A sin 2A = 2 sin AcosA 2 sinAcos B = sin(A + B) + sin(A − B)cos(A + B) = cosAcos B − sin Asin B cos 2 A = 1 (1 + cos2A) 2 sinAsin B = cos(A − B) − cos(A + B)2sin(A + B) = sin AcosB + cosAsin B sin 2 A = 1 (1 − cos2A) 2 cosAsin B = sin(A + B) − sin(A − B)2tan(A + B) =tan A + tanB1 − tan Atan Btan 2A =2 tanA1 − tan 2 Asin(−A) = − sinA cos(A − B) = cosAcos B + sin Asin B cos2A = 1 − tan2 A1 + tan 2 Atan(−A) = − tanA sin(A − B) = sin AcosB − cosAsin B sin 2A = 2 tanA1 + tan 2 Ae inθ = (cosθ + i sinθ) ntan A − tan Btan(A − B) == cosnθ + i sinnθ1 + tanAtan BLength/area/volumeTriangleParallelogramBcbCAhaahbCArea A = 1 2 ab sinC = 1 2 ah = √ s(s − a)(s − b)(s − c),where s = 1 2(a + b + c)aSine rule:sin A = bsin B = csinCCosine rule: a 2 = b 2 + c 2 − 2bc cosAArea A = ah = ab sinCTrapeziumRight-angled triangleabhcosA + cosB = 2 cos A + B2cosA − cosB = −2 sin A + B2sin A + sinB = 2 sin A + B2sin A − sinB = 2 cos A + B2Ac( a + b)Area A = h2bacos A − B2sin A − B2cos A − B2sin A − B2sin A = a ccosA = b ctan A = b cc 2 = a 2 + b 2CirclerlCircumference l = 2πrArea A = πr 2Arc/sectorθrlLength l = rθArea A = 1 2 r2 θ(θ in radians)rCylinderhCurved surface area A = 2πrhVolume V = πr 2 hConehrlCurved surface area A = πrlVolume V = 1 3 πr2 hrSphererSurface area A = 4πr 2Volume V = 4 3 πr3Frustrum of conehRlCurved surface area A = π(r + R)lVolume V = 1 3 πh(R2 + Rr + r 2 )