Maths Formula
Formulas (to differential equations) Math. A3, Midterm Test I.
sin2 x + cos2 x = 1 sin(x ± y) = sin x cos y ± cos x sin y tan(x ± y) = tan x±tan y 1∓tan x·tan y
differentiation rules: (cu) = cu
′ ′ ′ ′ ′
(c is constant)
cos(x ± y) = cos x cos y ∓ sin x sin y
(u + v) = u + v
(uv)′ = u′ v + uv ′
′ ′ u ′ = u v−uv v v2 df dg d dx f (g(x)) = dg dx
sin 2x = 2 sin x cos x tan 2x = sin x =
2
cos 2x = cos2 x − sin2 x
2 tan x 1−tan2 x 1−cos 2x , 2
integration rules: cos x =
2 1+cos 2x 2
cf dx = c
f dx f dx +
1 a F (ax
(c is constant) g dx + b) + c,
sin x + sin y = 2 sin x+y cos x−y 2 2 sin x − sin y = 2 cos x+y sin x−y 2 2 cos x + cos y = 2 cos x+y cos x−y 2 2 x−y x+y cos x − cos y = −2 sin 2 sin 2 1 sin x cos y = 2 [sin(x + y) + sin(x − y)] 1 cos x cos y = 2 [cos(x + y) + cos(x − y)] 1 sin x sin y = − 2 [cos(x + y) − cos(x − y)] x −x x −x cosh x = e +e sinh x = e −e , 2 2 2 2 cosh x − sinh x = 1 sinh 2x = 2 sinh x cosh x cosh 2x = cosh2 x + sinh2 x cosh2 x = cosh 2x+1 , 2
(f + g) dx =
f (ax + b) dx =
where F is antiderivative of f f (g(x))g ′ (x) dx = F (g(x)) + c, where F is antiderivative of f f f dx = f′ f ′ α ′ f α+1 α+1
uv dx = uv − R(e ) √ R( ax + b) x dx = ln |f | + c
+ c, if α = −1 u′ v dx ex = t √ ax + b = t
√ √ax+b cx+d
notable substitutions:
sinh2 x =
cosh 2x−1 2
R
√ √ax+b cx+d
=t
derivatives: (sinh x) = cosh x (cosh x) = sinh x (loga x) x ′ ′ ′ 1 = x ln a α−1 ′
R(sin x, cos x) √ R(x, a2 − x2 ) √ R(x, a2 + x2 ) √ R(x, x2 − a2 ) xα dx = α+1 sin x, cos x, tan x, tan x = t 2 x = a sin t, x = a cos t x = a sinh t x = a cosh t antiderivatives: (α = −1)
(xα )′ = αx (e ) = e x (ax )′ = ax ln(a) (sin x)′ = cos x
1 cos2 x 1 (cot x)′ = − sin2 x 1 (ln x)′ = x 1 (arc sin x)′ = √1−x2 1 (arc tan x)′ = 1+x2 1 (ar sinh x)′ = √1+x2 (ar cosh x)′ = √x1 −1 2 1 ′ (ar tanh x) = 1−x2 1 (ar coth x)′ = 1−x2 1 (arc cos x)′ = − √1−x2 1 (arc cot x)′ = − 1+x2
x α+1 + c 1 ax e dx = a eax
sin2 x + cos2 x = 1 sin(x ± y) = sin x cos y ± cos x sin y tan(x ± y) = tan x±tan y 1∓tan x·tan y
differentiation rules: (cu) = cu
′ ′ ′ ′ ′
(c is constant)
cos(x ± y) = cos x cos y ∓ sin x sin y
(u + v) = u + v
(uv)′ = u′ v + uv ′
′ ′ u ′ = u v−uv v v2 df dg d dx f (g(x)) = dg dx
sin 2x = 2 sin x cos x tan 2x = sin x =
2
cos 2x = cos2 x − sin2 x
2 tan x 1−tan2 x 1−cos 2x , 2
integration rules: cos x =
2 1+cos 2x 2
cf dx = c
f dx f dx +
1 a F (ax
(c is constant) g dx + b) + c,
sin x + sin y = 2 sin x+y cos x−y 2 2 sin x − sin y = 2 cos x+y sin x−y 2 2 cos x + cos y = 2 cos x+y cos x−y 2 2 x−y x+y cos x − cos y = −2 sin 2 sin 2 1 sin x cos y = 2 [sin(x + y) + sin(x − y)] 1 cos x cos y = 2 [cos(x + y) + cos(x − y)] 1 sin x sin y = − 2 [cos(x + y) − cos(x − y)] x −x x −x cosh x = e +e sinh x = e −e , 2 2 2 2 cosh x − sinh x = 1 sinh 2x = 2 sinh x cosh x cosh 2x = cosh2 x + sinh2 x cosh2 x = cosh 2x+1 , 2
(f + g) dx =
f (ax + b) dx =
where F is antiderivative of f f (g(x))g ′ (x) dx = F (g(x)) + c, where F is antiderivative of f f f dx = f′ f ′ α ′ f α+1 α+1
uv dx = uv − R(e ) √ R( ax + b) x dx = ln |f | + c
+ c, if α = −1 u′ v dx ex = t √ ax + b = t
√ √ax+b cx+d
notable substitutions:
sinh2 x =
cosh 2x−1 2
R
√ √ax+b cx+d
=t
derivatives: (sinh x) = cosh x (cosh x) = sinh x (loga x) x ′ ′ ′ 1 = x ln a α−1 ′
R(sin x, cos x) √ R(x, a2 − x2 ) √ R(x, a2 + x2 ) √ R(x, x2 − a2 ) xα dx = α+1 sin x, cos x, tan x, tan x = t 2 x = a sin t, x = a cos t x = a sinh t x = a cosh t antiderivatives: (α = −1)
(xα )′ = αx (e ) = e x (ax )′ = ax ln(a) (sin x)′ = cos x
1 cos2 x 1 (cot x)′ = − sin2 x 1 (ln x)′ = x 1 (arc sin x)′ = √1−x2 1 (arc tan x)′ = 1+x2 1 (ar sinh x)′ = √1+x2 (ar cosh x)′ = √x1 −1 2 1 ′ (ar tanh x) = 1−x2 1 (ar coth x)′ = 1−x2 1 (arc cos x)′ = − √1−x2 1 (arc cot x)′ = − 1+x2
x α+1 + c 1 ax e dx = a eax