Download SSCE 1793 LISTS OF FORMULAE Trigonometric cos2 x + sin2 x = 1

SSCE 1793
LISTS OF FORMULAE
Trigonometric
Hiperbolic
cos2 x + sin2 x = 1
sin 2x = 2 sin x cos x
ex − e−x
2
ex + e−x
cosh x =
2
cosh2 x − sinh2 x = 1
cos 2x = cos2 x − sin2 x
1 − tanh2 x = sech2 x
sinh x =
1 + tan2 x = sec2 x
cot2 x + 1 = cosec2 x
= 2 cos2 x − 1
coth2 x − 1 = cosech2 x
= 1 − 2 sin x
2 tan x
tan 2x =
1 − tan2 x
sin(x ± y) = sin x cos y ± cos x sin y
2
cos(x ± y) = cos x cos y ∓ sin x sin y
tan x ± tan y
tan(x ± y) =
1 ∓ tan x tan y
2 sin x cos y = sin(x + y) + sin(x − y)
2 sin x sin y = − cos(x + y) + cos(x − y)
2 cos x cos y = cos(x + y) + cos(x − y)
sinh 2x = 2 sinh x cosh x
cosh 2x = cosh2 x + sinh2 x
= 2 cosh2 x − 1
= 1 + 2 sinh2 x
2 tanh x
tanh 2x =
1 + tanh2 x
sinh(x ± y) = sinh x cosh y ± cosh x sinh y
cosh(x ± y) = cosh x cosh y ± sinh x sinh y
tanh x ± tanh y
tanh(x ± y) =
1 ± tanh x tanh y
Inverse Hiperbolic
Logarithm
sinh−1 x = ln(x +
ax = ex ln a
logb x
loga x =
logb a
√
x2 + 1), −∞ < x < ∞
√
cosh−1 x = ln(x + x2 − 1), x ≥ 1
(
)
1
1+x
−1
tanh x = ln
, −1 < x < 1
2
1−x
1
SSCE 1793
Differentiations
d
[k] = 0,
dx
Integrations
∫
k constant
d n
[x ] = nxn−1
dx
d
1
[ln |x|] =
dx
x
d
[cos x] = − sin x
dx
d
[sin x] = cos x
dx
d
[tan x] = sec2 x
dx
d
[cot x] = −cosec2 x
dx
d
[sec x] = sec x tan x
dx
d
[cosec x] = −cosec x cot x
dx
d x
[e ] = ex
dx
d
[cosh x] = sinh x
dx
d
[sinh x] = cosh x
dx
d
[tanh x] = sech2 x
dx
d
[coth x] = −cosech2 x
dx
d
[sech x] = −sech x tanh x
dx
d
[cosech x] = −cosech x coth x
dx
d
ln | sec x + tan x| = sec x
dx
d
ln |cosec x + cot x| = −cosec x
dx
kdx = kx + C
∫
xn dx =
∫
xn+1
+ C, n ̸= −1
n+1
dx
= ln |x| + C
x
∫
sin x dx = − cos x + C
∫
cos x dx = sin x + C
∫
sec2 x dx = tan x + C
∫
cosec2 x dx = − cot x + C
∫
sec x tan x dx = sec x + C
∫
cosec x cot x dx = −cosec x + C
∫
ex dx = ex + C
∫
sinh x dx = cosh x + C
∫
cosh x dx = sinh x + C
∫
sech2 x dx = tanh x + C
∫
cosech2 x dx = − coth x + C
∫
sech x tanh x dx = −sech x + C
∫
cosech x coth x dx = −cosech x + C
∫
sec x dx = ln | sec x + tan x| + C
∫
cosec x dx = − ln |cosec x + cot x| + C
2
SSCE 1793
Integrations Resulting
in Inverse Functions
Differentiations of
Inverse Functions
∫
d
1
du
[sin−1 u] = √
, |u| < 1.
·
dx
1 − u2 dx
√
d
−1
du
[cos−1 u] = √
·
, |u| < 1.
2
dx
1 − u dx
∫
d
1
du
[tan−1 u] =
.
·
2
dx
1 + u dx
∫
(x)
dx
= sin−1
+ C.
a
a2 − x2
a2
(x)
1
dx
= tan−1
+ C.
2
+x
a
a
(x)
1
dx
= sec−1
+ C.
a
a
|x| x2 − a2
√
−1
du
d
·
[cot−1 u] =
.
dx
1 + u2 dx
∫
d
1
du
√
[sec−1 u] =
, |u| > 1.
·
2
dx
|u| u − 1 dx
−1
du
d
√
·
[cosec−1 u] =
, |u| > 1.
2
dx
|u| u − 1 dx
∫
1
du
d
·
[cosh−1 u] = √
, |u| > 1.
2
dx
u − 1 dx
d
−1
du
√
[cosech−1 u] =
·
, u ̸= 0.
2
dx
|u| 1 + u dx
3
√
(x)
dx
+ C, x > 0.
= cosh−1
a
x2 − a2
dx
a2 − x2
 1
( )
−1 x

+ C,
 tanh
a
a
=
( )

 1 coth−1 x + C,
a
a
|x| < a,
|x| > a.
∫
(x)
dx
1
√
= − sech−1
+ C,
a
a
x a2 − x2
0 < x < a.
∫
x
dx
1
√
= − cosech−1 + C,
2
2
a
a
x a +x
0 < x < a.
d
1
du
[coth−1 u] =
·
, |u| > 1.
dx
1 − u2 dx
−1
d
du
[sech−1 u] = √
, 0 < u < 1.
·
2
dx
u 1 − u dx
(x)
dx
+ C, a > 0.
= sinh−1
a
x2 + a2
∫
d
1
du
[sinh−1 u] = √
·
dx
u2 + 1 dx
d
1
du
[tanh−1 u] =
·
, |u| < 1.
dx
1 − u2 dx
√