Summary Of Important Laws Of Differentiation And Integration Mahmoud El Shafee , Belal El Khatib

1. Summary Of Important Laws Of Differentiation
And Integration
Mahmoud El Shafee , Belal El Khatib
1 Relations
1.1 Trigonometry
Figure 1: Geometrical Definitions of Sine and Cosine
sin2
(θ) + cos2
(θ) = 1 (1)
1 + tan2
(θ) = sec2
(θ) (2)
1 + cot2
(θ) = csc2
(θ) (3)
sin(2θ) = 2sin(θ)cos(θ) (4)
cos(2θ) = cos2
(θ) − sin2
(θ)
= 2cos2
(θ) − 1
= 1 − 2sin2
(θ)
(5)
sin2
(θ) =
1
2
(1 − cos(2θ)) (6)
1

2. cos2
(θ) =
1
2
(1 + cos(2θ)) (7)
sin(θ ± φ) = sin(θ)cos(φ) ± cos(θ)sin(φ) (8)
cos(θ ± φ) = cos(θ)cos(φ) sin(θ)sin(φ) (9)
sin(θ)cos(φ) =
1
2
(sin(θ + φ) + sin(θ − φ)) (10)
cos(θ)cos(φ) =
1
2
(cos(θ + φ) + cos(θ − φ)) (11)
sin(θ)sin(φ) =
1
2
(cos(θ − φ) − cos(θ + φ)) (12)
sec(θ) =
1
cos(θ)
csc(θ) =
1
sin(θ)
cot(θ) =
1
tan(θ)
=
cos(θ)
sin(θ)
(13)
sin(θ) =
Opp
Hyp
cos(θ) =
Adj
Hyp
tan(θ) =
Opp
Adj
(14)
1.2 Hyper Trignometry
sinh(x) =
ex
− e−x
2
(15)
cosh(x) =
ex
+ e−x
2
(16)
tanh(x) =
sinh(x)
cosh(x)
=
ex
− e−x
ex + e−x
(17)
sech(x) =
1
cosh(x)
=
2
ex + e−x
(18)
csch(x) =
1
sinh(x)
=
2
ex − e−x
(19)
coth(x) =
1
tanh(x)
=
cosh(x)
sinh(x)
=
ex
+ e−x
ex − e−x
(20)
cosh2
(x) − sinh2
(x) = 1 (21)
1 − tanh2
(x) = sech2
(x) (22)
2

3. coth2
(x) − 1 = csch2
(x) (23)
1
arcsinh(x) = ln(x +
√
x2 + 1) (24)
arccosh(x) = ln(x +
√
x2 − 1) (25)
arctanh(x) =
1
2
ln(
1 + x
1 − x
) (26)
sin(−θ) = −sin(θ)
cos(−θ) = cos(θ)
(27)
sinh(−θ) = −sinh(θ)
cosh(−θ) = cosh(θ)
(28)
1.3 Exponential and Logarithms
ax
= y ⇐⇒ x = loga(y) (29)
ex
= y ⇐⇒ x = ln(y) (30)
logB(A) =
lnA
lnB
(31)
eln ∎
= ∎,ln(e∎
) = ∎ (32)
lnAB = lnA + lnB (33)
ln
A
B
= lnA − lnB (34)
lnAn
= nlnA (35)
e∞
= ∞ ⇒ ln∞ = ∞ (36)
e−∞
= 0 ⇒ ln0 = −∞ (37)
e0
= 1 ⇒ ln1 = 0 (38)
lne = 1 (39)
a∞
=
⎧⎪⎪
⎨
⎪⎪⎩
0 0 < a < 1
∞ a > 1
(40)
∞
∎
= ∞,
∎
∞
= 0, ∎ × ∞ = ∞ (41)
0
∎
= 0,
∎
0
= ∞, ∎ × 0 = 0 (42)
∎0
= 1 (43)
1arcsinh is the inverse function of sinh, or it may be written as sinh−1. The same for the
other trigonometric and hyper trigonometric functions.
3

4. 2 Differentiation Summary
d
dx
(c) = 0 (44)
d
dx
(x) = 1 (45)
d
dx
(xn
) = nxn−1
(46)
d
dx
(∎n
) = n ∎n−1
∎′
(47)
d
dx
(ex
) = ex
(48)
d
dx
(e∎
) = ∎′
e∎
(49)
d
dx
(ax
) = ax
ln(a) (50)
d
dx
(a∎
) = a∎
∎′
ln(a) (51)
d
dx
(ln(x)) =
1
x
(52)
d
dx
(loga(x)) =
1
ln(a)
1
x
(53)
d
dx
(loga(∎)) =
1
ln(a)
∎′
∎
(54)
d
dx
(sin(x)) = cos(x) (55)
d
dx
(cos(x)) = −sin(x) (56)
d
dx
(tan(x)) = sec2
(x) (57)
d
dx
(cot(x)) = −csc2
(x) (58)
d
dx
(sec(x)) = sec(x)tan(x) (59)
d
dx
(csc(x)) = −csc(x)cot(x) (60)
d
dx
(arcsin(x)) =
1
√
1 − x2
(61)
d
dx
(arccos(x)) =
−1
√
1 − x2
(62)
4

5. d
dx
(arctan(x)) =
1
1 + x2
(63)
d
dx
(arccot(x)) =
−1
1 + x2
(64)
d
dx
(arcsec(x)) =
1
x
√
x2 − 1
(65)
d
dx
(arccsc(x)) =
1
x
√
x2 − 1
(66)
d
dx
(sinh(x)) = cosh(x) (67)
d
dx
(cosh(x)) = sinh(x) (68)
d
dx
(tanh(x)) = sech2
(x) (69)
d
dx
(coth(x)) = −csch2
(x) (70)
d
dx
(sech(x)) = −sech(x)tanh(x) (71)
d
dx
(csch(x)) = −csch(x)coth(x) (72)
d
dx
(arcsinh(x)) =
1
√
x2 + 1
(73)
d
dx
(arccosh(x)) =
1
√
x2 − 1
(74)
d
dx
(arctanh(x)) =
1
1 − x2
(75)
d
dx
(arccoth(x)) =
1
1 − x2
(76)
d
dx
(arcsech(x)) =
−1
x
√
1 − x2
(77)
d
dx
(arccsch(x)) =
−1
x
√
1 + x2
(78)
Product rule
(fg)′
= fg′
+ f′
g (79)
Quotient rule
(
f
g
)
′
=
gf′
− fg′
g2
(80)
5

6. 3 Integration Summary
∫ cdx = cx + const (81)
∫ xdx =
1
2
x2
+ c (82)
∫ xn
dx =
1
n + 1
xn+1
+ c , n ≠ −1 (83)
∫
1
x
dx = ln(x) + c (84)
∫ ∎′
∎n
dx =
1
n + 1
∎n+1
+c (85)
∫
∎′
∎
dx = ln(∎) + c (86)
∫ ex
dx = ex
+ c (87)
∫ ∎′
e∎
dx = e∎
+ c (88)
∫ ax
dx =
1
ln(a)
ax
+ c (89)
∫ ∎′
cos(∎)dx = sin(∎) + c (90)
∫ ∎′
sin(∎)dx = −cos(∎) + c (91)
∫ ∎′
sec2
(∎)dx = tan(∎) + c (92)
∫ ∎′
csc2
(∎)dx = −cot(∎) + c (93)
2
∫
1
√
1 − x2
dx = arcsin(x) + c = −arccos(x) + c′
(94)
∫
1
√
a2 − x2
dx = arcsin(
x
a
) + c (95)
∫
∎′
√
1 − ∎2
dx = arcsin(∎) + c (96)
∫
1
1 + x2
dx = arctan(x) + c (97)
∫
1
a2 + x2
dx =
1
a
arctan(
x
a
) + c (98)
∫
∎′
1 + ∎2
dx = arctan(∎) + c (99)
2c′ = π
2
+ c
6

7. ∫
1
x
√
x2 − 1
dx = arcsec(x) + c (100)
∫
1
x
√
x2 − a2
dx =
1
a
arcsec(
x
a
) + c (101)
∫
∎′
∎
√
∎2 − 1
dx = arcsec(∎) + c (102)
∫
1
√
1 + x2
dx = arcsinh(x) + c (103)
∫
1
√
x2 − 1
dx = arccosh(x) + c (104)
∫
1
1 − x2
dx = arctanh(x) + c (105)
∫
1
x
√
1 − x2
dx = −arcsech(x) + c (106)
∫
1
x
√
1 + x2
dx = −arccsch(x) + c (107)
∫ sec(x)dx = ln(sec(x) + tan(x)) + c (108)
∫ csc(x)dx = ln(csc(x) − cot(x)) + c (109)
∫ tan(x)dx = ln(sec(x)) + c (110)
∫ cot(x)dx = ln(sin(x)) + c (111)
Integral by Parts rule
∫ udv = uv − ∫ vdu (112)
7

8. 4 Some Important Graphs
4.1 Exponential and Logarithmic Functions
−5 5 10
−5
5
10
f(x) = ex
f(x) = ln(x)
4.2 Trigonometric Functions
−2π −3
2
π −π −π
2
π
2
π 3
2
π 2π
−1
−0.5
0.5
1
f(x) = sinx
f(x) = cosx
8

9. −3
2
π −π −π
2
π
2
π 3
2
π
−10
−5
5
10
f(x) = tanx
4.3 Reciprocal
−10 −5 5 10
−10
−5
5
10
f(x) = x
f(x) = 1
x
9

10. 4.4 Square and Square roots
−4 −2 2 4
−4
−2
2
4
f(x) = x2
f(x) =
√
x
f(x) = −
√
x
4.5 Cube and Cubic root
−4 −2 2 4
−4
−2
2
4 f(x) = x3
f(x) = 3
√
x
f(x) = 3
√
x
10