UCC Mathematical Tables Calculus f (x) xn ln |x| cos x sin x tan x sec x cosec x cot x ex eax ax x cos−1 a −1 x sin a x tan−1 a x sec−1 a −1 x cosec a Product rule Quotient rule Chain rule Integration by parts Trapezoidal rule f ′ (x) nxn−1 1 x − sin x cos x sec2 x sec x tan x − cosec x cot x − cosec2 x ex aeax ax ln a 1 −√ a2 − x2 1 √ 2 a − x2 a a2 + x2 a √ x x2 − a2 a − √ 2 x x − a2 f (x) −1 cot x a sinh x cosh x tanh x sech x cosech x coth x sinh−1 x cosh−1 x tanh−1 x sech−1 x cosech−1 x coth−1 x f ′ (x) f (x) a − 2 a + x2 cosh x sinh x sech2 x − sech x tanh x − cosech x coth x − cosech2 x 1 √ 2 x +1 1 √ x2 − 1 1 1 − x2 1 − √ x 1 − x2 1 − √ x x2 + 1 1 − 2 x −1 dy dv du =u +v dx dx dx dv du v −u u dy dx dx y= ⇒ = v dx v2 du dv f (x) = u v(x) ⇒ f ′ (x) = dv dx Z Z u(x)v ′ (x) dx = uv − v(x)u′ (x) dx y = uv ⇒ y1 Taylor series (centre a) Maclaurin series Volume of solid of revolution about x-axis Simpson’s rule (n odd) y2 h y3 f (x) dx xn+1 xn (n 6= −1) n+1 1 ln |x| x cos x sin x sin x − cos x tan x ln | sec x| sec x ln | sec x + tan x| x cosec x lntan 2 cot x ln | sin x| ex ex 1 ax eax e ax a ax ln a x 1 √ sin−1 2 2 a a −x 1 1 −1 x tan x2 + a2 a a 1 1 x −1 √ sec a a x x2 − a2 Newton-Raphson i hh A≈ y1 + yn + 2(y2 + y3 + · · · + yn−1 ) 2 R y4 xn+1 = xn − f (x) 1 √ x2 + a2 1 2 a − x2 1 √ x2 − a2 sinh x cosh x tanh x coth x sech x cosech x cos2 x sin2 x cosh2 x sinh2 x 1 √ 2 x a − x2 1 √ 2 x x + a2 R f (x) dx x + √x2 + a2 ln a 1 a + x ln 2a a−x x + √x2 − a2 ln a cosh x sinh x ln cosh x ln | sinh x| −1 tan x) (sinh x lntanh 2 1 1 2 (x + 2 sin 2x) 1 1 2 (x − 2 sin 2x) 1 1 2 (x + 2 sinh 2x) 1 1 2 (−x + 2 sinh 2x) 1 x − sech−1 a a 1 −1 x − cosech a a f (xn ) f ′ (xn ) f ′′ (a) 2 f (r) (a) r x + ···+ x + ··· 2! r! f ′′ (0) 2 f (r) (0) r f (x) = f (0) + f ′ (0)x + x + ···+ x + ··· 2! r! Z x=b V = πy 2 dx f (a + x) = f (a) + f ′ (a)x + x=a i hh A≈ y1 + yn + 2(y3 + y5 + · · · + yn−2 ) + 4(y2 + y4 + · · · + yn−1 ) 3 yn Trigonometry sin A cos A cos A 1 cot A = = sin A tan A 1 sec A = cos A 1 cosec A = sin A tan A = cos2 A + sin2 A = 1 cos 2A = cos2 A − sin2 A 2 cos A cos B = cos(A + B) + cos(A − B) sec2 A = 1 + tan2 A sin 2A = 2 sin A cos A 2 sin A cos B = sin(A + B) + sin(A − B) cos(A + B) = cos A cos B − sin A sin B sin(A + B) = sin A cos B + cos A sin B tan A + tan B 1 − tan A tan B cos(−A) = cos A tan(A + B) = sin(−A) = − sin A cos(A − B) = cos A cos B + sin A sin B tan(−A) = − tan A sin(A − B) = sin A cos B − cos A sin B einθ = (cos θ + i sin θ)n tan(A − B) = = cos nθ + i sin nθ 1 (1 + cos 2A) 2 1 sin2 A = (1 − cos 2A) 2 2 tan A tan 2A = 1 − tan2 A 1 − tan2 A cos 2A = 1 + tan2 A 2 tan A sin 2A = 1 + tan2 A cos2 A = 2 sin A sin B = cos(A − B) − cos(A + B) 2 cos A sin B = sin(A + B) − sin(A − B) A+B A−B cos 2 2 A+B A−B cos A − cos B = −2 sin sin 2 2 A+B A−B sin A + sin B = 2 sin cos 2 2 A+B A−B sin A − sin B = 2 cos sin 2 2 cos A + cos B = 2 cos tan A − tan B 1 + tan A tan B Length/area/volume Triangle A c b h C B a p Area A = 12 ab sin C = 12 ah = s(s − a)(s − b)(s − c), where s = 12 (a + b + c) a b c Sine rule: = = sin A sin B sin C Cosine rule: a2 = b2 + c2 − 2bc cos A Right-angled triangle a c b cos A = c a tan A = b c2 = a 2 + b 2 sin A = c a A b b Parallelogram b h Area A = ah = ab sin C Trapezium h Area A = a + b h 2 C a a l Circle r Circumference l = 2πr Area A = πr2 Arc/sector Curved surface area A = 2πrh Volume V = πr2 h Cone θ Length l = rθ Area A = 12 r2 θ r l (θ in radians) r Cylinder h l h Curved surface area A = πrl Volume V = 13 πr2 h r r Sphere r Surface area A = 4πr Volume V = 43 πr3 2 Frustrum of cone l h R Curved surface area A = π(r + R)l Volume V = 13 πh(R2 + Rr + r2 )