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Санкт-Петербургский государственный электротехнический университет "ЛЭТИ"

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Электротехника

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(EOD).Reference information.pdf

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<<< < Предыдущая 3 4 5 6 7 8 9 10 11 12 13 1415 / 4615 16 17 18 19 20 21 22 23 24 25 26 27 > Следующая >>>

page 58

2.6.1 Single Variable Functions

2.6.1.1 - Differentiation

• The basic principles of differentiation are,

page 59

Both u, v and w are functions of x, but this is not shown for brevity.

Also note that C is used as a constant, and all angles are in radians.

(C ) = 0

-----

(u )

-----

(Cu ) = (C )-----

(u + v + …

) =

…

-----

-----(u )+ -----(v )+

n – 1

-----(u

) =

(nu

)-----(u )

-----

(uv ) = (u )-----(v )+

(v )-----(u )

-----

----

-----(u )–

----

-----(v )

(uvw ) =

(u )

-----

(uv )-----(w )+ (uw )-----

(v )+ (vw )-----

(y ) =

(u ) = chain rule

-----

(y )-----

(u ) =

-----

-------------

(x )

-----

(y )

-----

(y ) =

-----

-------------

(x )

-----

• Differentiation rules specific to basic trigonometry and logarithm functions

page 60

d		(sin u )			=				d
-----		(sin u )			=		(cos u )-----(u )
dx									dx
d		(cos u ) =							d	(u )
-----		(cos u ) =						(– sin u )-----		(u )
dx									dx
d		(tan u )			=		1		2 d	(u )
-----		(tan u )			=			-----------	-----	(u )
dx							cos u		dx
d	(e		u	) = (e		u		d
-----	(e			) = (e				)-----(u )
dx								dx
d	(ln x ) =				1
-----	(ln x ) =				--
dx					x

d		2	d		(u )
-----	(cot u ) = (– csc u )		-----		(u )
dx			dx
d					d
-----	(sec u ) = (tan u sec u )-----(u )
dx					dx
d					d
-----	(csc u ) = (– csc u cot u )-----(u )
dx					dx
d	(sinh u ) =		d		(u )
-----	(sinh u ) =	(cosh u )-----			(u )
dx			dx
d	(cosh u ) =		d		(u )
-----	(cosh u ) =	(sinh u )-----			(u )
dx			dx
d	(tanh u ) =		2	d
-----	(tanh u ) =	(sech u )		-----(u )
dx				dx

• L’Hospital’s rule can be used when evaluating limits that go to infinity.

lim	f(x )	=
	----------
x → a
	g(x )

d	f(x )
----	f(x )

dt
lim ---------------------		=
x → a d	g(x )
----	g(x )
dt

d	2
----	f(x )

dt			…
lim -----------------------		=	…
x → a d	2
----	g(x )
dt

• Some techniques used for finding derivatives are,

Leibnitz’s Rule, (notice the form is similar to the binomial equation) can be used for finding the derivatives of multiplied functions.

n – 1

(uv ) =

(v )

-----

(u ) -----

(v )+

-----

(u ) -----

n d

n – 2

(v )+

…

n d

(v )

-----

(u ) -----

-----

(u )

-----

page 61

2.6.1.2 - Integration

• Some basic properties of integrals include,

In the following expressions, u, v, and w are functions of x. in addition to this, C is a constant. and all angles are radians.

∫

Cdx = ax + C

∫

Cf(x )dx = C f(x )dx

∫

(u + v + w + … )dx = udx + vdx + wdx + …

∫

uv –

∫

udv =

vdu = integration by parts

∫

u =

f(Cx )dx =

---

f(u )du

F(f(x ))dx =

(x )du =

F(u )

u = f(x )

F(u )-----

-----------du

∫

∫f'(x )

∫

xn + 1

-----------

+ C

∫

+ C

x dx =

--dx =

n + 1

dx =

+ C

dx =

+ C

-------

∫

ln a

∫

• Some of the trigonometric integrals are,

page 62

sin xdx =

– cos x + C

sin 2 x

sin 4x

+ C

(cos x ) dx

= -----

-------------4

-------------32

∫

n + 1

∫

sin x + C

∫

(sin x )

+ C

n + 1

cos xdx =

cos x(sin x ) dx =

------------------------

∫

sin x cos x + x

+ C

∫

cosh x + C

(sin x ) dx = –

sinh xdx =

∫

sin x cos x + x

+ C

∫

sinh x + C

(cos x ) dx =

cosh xdx =

∫

cos x (( sin x ) + 2 )

∫

ln (cosh x )+ C

+ C

(sin x ) dx = – -------------------------------------------

tanh xdx =

∫

sin x (( cos x ) + 2 )

(cos x ) dx = -------------------------------------------

x cos (ax )dx ------------------- --=

cos ( ax ) + x sin (ax )+ C

∫

cos (ax )dx =

2 x cos ( ax )

a2x2

– 2

sin (ax )+ C

∫

• Some other integrals of use that are basically functions of x are,

page 63

xn + 1

---

----- + C

∫

n + 1

∫

–1

ln (a + bx )

+ C

(a + bx ) dx =

------

-----

--------

–1

a + 2 –b

+ C,a >0,b <0

(a + bx2 )

dx = ----

-----

-------

∫

(–b )a

a – x –b

∫

–1

ln (bx2 + a )

(a + bx

) dx =

+ C

-----

---------

---

∫

–1

(a + bx

) dx =

–

--------

---

atan -

----

-----

+ C

–1

a + x

+ C,a

(a – x )

dx = ----- ln --------

---

∫

a – x

∫

–1

a + bx

+ C

(a + bx ) dx =

------

-----

–1--

x(x2

± a2 ) 2 dx = x2 ± a2 + C

∫

(a + bx + cx2 ) dx =

1---- ln a + bx + cx2 + x c + b--------

+ C,c >0

∫

–1

asin

– 2cx – b

+ C,c <0

(a + bx + cx2 )

dx =

∫

–c

– 4ac

page 64

							1													3
							--				2									--
∫							2				2									2
(a + bx ) dx = -----(a + bx )
											3b
							1													3
							--				2									--
∫							2				2									2
(a + bx ) dx = -----(a + bx )
											3b
									1																									3
									1																									--
									--																									2
									2			2(2a – 3bx )(a + bx )
x(a + bx ) dx = –----------------------------------------------------
∫																				15b2
																																												1
																																					1							--
																													ln								1					2 2
																								1					ln			x +				---- + x
																								1														2
																								--												a
									1			x(1 + a							2	x	2			2												a
									--			x(1 + a								x				) +					--------------------------------------------
∫					2		2		2																				2								a
∫									) dx =																				2
(1 + a x									) dx =			----------------------------------------------------------------------------------
																									3
														1											--
											1			1				+ x				2		2
											1	a			----			+ x
											--			a2
∫						2			2		2			a2				3
∫	(1 + a x ) dx =																	3
x	(1 + a x ) dx =											----------------------------
																																																					1
																																																	1				--
																																																	1		+ x	2	2
											1																	3																		1		ln x +	----		+ x
											1																	3																		1			2
											--		ax					1										--			8															--			a
x	2	(1 + a					2		x	2 2			ax					1		+ x						2		2			8										2		x	2		2			a
x		(1 + a							x		) dx = -----					----				+ x									– --------x(1 + a														x		)		– --------------------------------------------
∫												4							2												8a			2															8a	3
																	a														8a																		8a
									1																			1
									--		1																	--			asin (ax )
∫				2			2		2		1										2				2			2			asin (ax )
∫									) dx = --				x(1 – a x ) + --------------------
(1 – a x									) dx = --				x(1 – a x ) + --------------------
											2																								a
											1																	3
											1																	3
											--			a	1													--
x(1 – a						2	x		2		2			a	1				– x					2 2
x(1 – a							x			) dx = –--						----			– x
∫												3						2
																a
										1																	3														1
										--				x													--		1												--							x
x	2	(a	2		– x			2		2				x	(a			2	– x					2			2		1				x(a			2	– x		2		2						2	x
x		(a			– x					) dx =		– --			(a				– x							) + --							x(a				– x			)			+ a					asin --
∫												4																	8																			a

									–		1																					1
											1																					1
											--	1												1							--
(1 + a					2	x	2		)		2	1				x +								1				+ x		2	2
(1 + a						x			)		dx = -- ln					x +							----					+ x
∫												a													2
																							a
									–		1
									–		--	1																	1
∫				2			2				2	1																	1
∫									)		dx =	-- asin (ax ) = –-- acos (ax )
(1 – a x									)		dx =	-- asin (ax ) = –-- acos (ax )
												a																	a

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