????????

admin

?????????????

?????

(tgx)??sec2x(arcsinx)??

1

(ctgx)???csc2x?x2

(secx)??secx?tgx(arccosx)???

1

(cscx)???cscx?ctgx?x2(ax)??axlna(arctgx)??

1

1?x2

(log1ax)??

xlna

(arcctgx)???

1

1?x2

??????

?tgxdx??lncosx?CC??dxctgxdx?lnsinx?C

cos2x??sec2

xdx?tgx??secxdx?lnsecx?tgx?C?dx2

sin2x??cscxdx??ctgx?C

?cscxdx?lncscx?ctgx?C

?secx?tgxdx?secx?C

?dx?cscx?ctgxdx??cscx?C

a2?x2?1aarctgx

a?C?dx1x??ax

dx?ax

lna?C

x2?a2?2alna

x?a?C?shxdx?chx?C?dxa2?x2?1a?x

2alna?x?C?chxdx?shx?C?dxa2?x2

?arcsinx

a

?C?

dxx2?a2

?ln(x?x2?a2)?C

?

?

2

2

In

n??sinxdx??cosnxdx?

n?1

n

In?2?x2

?a2

dx?x22

a22x?a?2ln(x?x2?a2)?C

?x?adx?x22222

a2x?a?2lnx?x2?a2?C

?

a2?x2dx?x22

a2x2a?x?2arcsina

?C

???????????

sinx?2u1?u2x2du

1?u2cosx?1?u2u?tg2??dx?1?u2

??????? ???????

x?x

????:shx?e?e2limsinx x?0x

?1 chx?

ex?e?x

????:2

limx??(1?1

x

)x?e?2.718281828459045...????:thx?shxex?e?x

chx?ex

?e?x arshx?ln(x?x2?1? archx??ln(x?x2?1) arthx?11?x

2ln

1?x

??????? �?????

�?????? �???????

sin(???)?sin?cos??cos?sin?sin??sin??2sin

???

cos(???)?cos?cos??sin?sin?2cos

???

2tg(???)?

tg??tg?

sin??sin??2cos??????

1?tg??tg?2sin

2cos??cos??2cos??????

ctg(???)?

ctg??ctg??1

2cos

2ctg??ctg?

cos??cos??2sin??????

2sin

2

�?????

sin2??2sin?cos?

cos2??2cos2??1?1?2sin2??cos2??sin2?sin3??3sin??4sin3?ctg2??

ctg2??1

cos3??4cos3??3cos?2ctg?tg3??

3tg??tg3?tg2??

2tg?

1?3tg2?

1?tg2?

�?????

sin?

?cos?12

??2????????????cos2???cos2

tg

?

??

1?cos?1?cos?sin??1?cos?1?cos?2

1?cos??sin??1?cos???ctg2??1?cos??sin??

sin?

1?cos?

�?????a?bsinB?c

sinC

?2R�?????c2sinA?a2?b2?2abcosC

�????????arcsinx?

?

2

?arccosx???arctgx?

?

2

?arcctgx

??????��?????Leibniz????

(uv)

(n)

n

??Cku(n?k)v(k)

nk?0

?u(n)v?nu(n?1)v??

n(n?1)(n?2)n(n?1)?(n?k?2!uv?????1)(n?k)(k)

k!

uv???uv(n)

??????????

?????????f(b)?f(a)?f?(?)(b?a)f(b)?f(a)f?(?)

F(b)?F(a)?

F?(?)

?F(x)?x???????????????????

???

??????ds??y?2dx,??y??tg??????K?

??

?s

??:?M??M???????????????s?MM????M?????K??lim??s?0?s?d?

ds?y??(1?y?2)

3.

???K?0;???a???K?1

a

.

?????????

b

?????f(x)?

b?a

n

(y0?y1???yn?1)ab

?????f(x)?

b?aa

n[1

2

(y0?yn)?y1???yn?1]

b

??????f(x)?

b?a

a

3n

[(y0?yn)?2(y2?y4???yn?2)?4(y1?y3???yn?1)]??????????

??W?F?s????F?p?A

???F?k

m1m2

r

2,k????? ???????y?1

bb?a?f(x)dxab

1b?a?f2(t)dta

????????????

??2?????d?M1M2?(x2?x1)2?(y2?y221)?(z2?z1)?????????Prju?cos?,???u?????

Prj??a???u(a12)?Prja1?Prjaa??b??a??b?

2cos??axbx?ayby?azbz,?????,?????????cos??axbx?ayby?azbz

a2

2

2

2

x?ay?az?bx?b2

2

y?bz

i

jk

c??a?

?b??ax

aya,c??a??b?sin?.??????v??w??r?z.bx

by

bz

ay???????[a?b?ax

azc?]?(a??b?)?c?

?bx

byba??b??c?

z?cos?,??????

cx

cy

cz

???????????

??????

1?????A(x?x(y?y???n?

0)?B0)?C(z?z0)?0?{A,B,C},M0(x0,y0,z0)2??????Ax?By?Cz?D?03xyz

a?b?c

?1

???????????????d?

Ax0?By0?Cz0?D

A2?B2?C2

x?x0m?y?y?x?x0?mt

0n?z?z0p?t,???s?{m,n,p};??????

?y?y?0?nt

?

z?z0?pt

?????

x2y2z2

1a2?b2?c2?1

x2y2

22p?2q?z?,p,q???

3?????

x2y2z2

a2?b2?c2?1

x2y2z2

a2?b2?c

2??????1

??????????

????dz?

?z?xdx??z?ydy???du??u?u?u?xdx??ydy??z

dz??????????z?dz?fx(x,y)?x?fy(x,y)?y???????????

z?f[u(t),v(t)]dz?z?u?z?v

dt??u??t??v??t

?

z?f[u(x,y),v(x,y)]?z?z?u?z?v

?x??u??x??v?

?x

?u?u(x,y)?v?v(x,y)??du?

?u?xdx??u?ydy???dv??v?xdx??v

?y

dy??????????

???F(x,y)?0dyFxd2y?F?

Fdydx??Fx2?(??(?x?

ydx?xFy?yFydx???F(x,y,z)?0?zFx?z

Fy?x??F??

z?yFz

???????????

????????

??????????f(-x)=-f(x)???????y???

?????

(tgx)??secx(ctgx)???cscx(secx)??secx?tgx(cscx)???cscx?ctgx(ax)??axlna

1

(logax)??

xlna

??????

2

2

(arcsinx)??

1

?x2

1

(arccosx)???

?x21

(arctgx)??

1?x2

1

(arcctgx)???

1?x2

?tgxdx??lncosx?C?ctgxdx?lnsinx?C

?secxdx?lnsecx?tgx?C?cscxdx?lncscx?ctgx?C

dx1x

?arctg?C?a2?x2aadx1x?a

?ln?x2?a22ax?a?Cdx1a?x

??a2?x22alna?x?Cdxx

?arcsin?C?a2?x2

a

?

2

n

dx2

?sec?cos2x?xdx?tgx?Cdx2

?csc?sin2x?xdx??ctgx?C

?secx?tgxdx?secx?C

?cscx?ctgxdx??cscx?C

ax

?adx?lna?C

x

?shxdx?chx?C?chxdx?shx?C?

dxx2?a2

?ln(x?x2?a2)?C

?

2

In??sinxdx??cosnxdx?

n?1

In?2n

???

x2a22

x?adx?x?a?ln(x?x2?a2)?C

22x2a2222

x?adx?x?a?lnx?x2?a2?C

22x2a2x222

a?xdx?a?x?arcsin?C

22a

2

2

???????????

2u1?u2x2du

sinx???cosx???u?tg??dx?

21?u21?u21?u2

??????????????

ex?e?x

????:shx?

2ex?e?x

????:chx?

2

shxex?e?x

????:thx??x

chxe?e?xarshx?ln(x?x2?1?archx??ln(x?x2?1)

11?x

arthx?ln

21?x

??????? �?????

sinx lim?1x?0x

1

lim(1?)x?e?2.718281828459045...x?? x

�?????? �???????

sin(???)?sin?cos??cos?sin?cos(???)?cos?cos??sin?sin?tg(???)?

tg??tg?1?tg??tg?ctg??ctg??1

ctg(???)?

ctg??ctg?

sin??sin??2sin

???

22??????

sin??sin??2cossin

22??????

cos??cos??2coscos

22??????

cos??cos??2sinsin

22

cos

???

�?????

sin2??2sin?cos?

cos2??2cos2??1?1?2sin2??cos2??sin2?ctg2??1

ctg2??

2ctg?2tg?

tg2??

1?tg2?

�?????

sin3??3sin??4sin3?cos3??4cos3??3cos?3tg??tg3?tg3??

1?3tg2?

sintg

?

2

????

?cos???cos?

????????????cos??222

1?cos1?cos?sin???cos1?cos?sin?

????ctg????

1?cos?sin?1?cos?21?cos?sin?1?cos?

abc

???2R �?????c2?a2?b2?2abcosC sinAsinBsinC

?

2

�?????

�????????arcsinx?

?

2

?arccosx???arctgx?

?

2

?arcctgx

??????��?????Leibniz????

(uv)

(n)

k(n?k)(k)

??Cnuvk?0

n

?u(n)v?nu(n?1)v??

n(n?1)(n?2)n(n?1)?(n?k?1)(n?k)(k)

uv?????uv???uv(n)

2!k!

??????????

?????????f(b)?f(a)?f?(?)(b?a)f(b)?f(a)f?(?)

?

F(b)?F(a)F?(?)

???

?F(x)?x???????????????????

??????ds??y?2dx,??y??tg??????K?

??

??:?M??M???????????????s?MM?????s

y????d?

M?????K?lim??.

23?s?0?sds(1?y?)

???K?0;1

???a???K?.

a

?????????

b

?????f(x)?

ab

b?a

(y0?y1???yn?1)n

b?a1

[(y0?yn)?y1???yn?1]n2

b?a

[(y0?yn)?2(y2?y4???yn?2)?4(y1?y3???yn?1)]3n

?????f(x)?

a

b

??????f(x)?

a

??????????

??W?F?s

????F?p?A

mm

???F?k122,k?????

r

b1

???????y?f(x)dx?b?aa1f2(t)dt?b?aa

????????????

b

??2?????d?M1M2?(x2?x1)2?(y2?y1)2?(z2?z1)2?????????Prju?cos?,??u?????

????Prju(a1?a2)?Prja1?Prja2????

a?b?a?bcos??axbx?ayby?azbz,?????,?????????cos??i

???

c?a?b?ax

bx

jayby

axbx?ayby?azbz

ax?ay?az?bx?by?bz

2

2

2

2

2

2

k

??????az,c?a?bsin?.??????v?w?r.bz

aybycy

az

???

bz?a?b?ccos?,??????

cz

ax

??????

???????[abc]?(a?b)?c?bx

cx???????????

??????

?

1?????A(x?x0)?B(y?y0)?C(z?z0)?0???n?{A,B,C},M0(x0,y0,z0)2??????Ax?By?Cz?D?0

xyz

3???1

abc???????????????d?

Ax0?By0?Cz0?D

A2?B2?C2

?x?x0?mt

x?xy?y0z?z0??

0???t,??s?{m,n,p};??????y?y0?nt

mnp?z?z?pt

0??????

x2y2z2

12?2?2?1

abcx2y2

2??z?,p,q???

2p2q3?????

x2y2z2

2?2?2?1

abcx2y2z2

2?2?2??????1

abc

??????????

????dz?

?z?z?u?u?udx?dy???du?dx?dy?dz?x?y?x?y?z

??????????z?dz?fx(x,y)?x?fy(x,y)?y???????????

dz?z?u?z?v

z?f[u(t),v(t)]?????

dt?u?t?v?t

?z?z?u?z?v

z?f[u(x,y),v(x,y)]????

?x?u?x?v?x

?u?u(x,y)?v?v(x,y)??du?

?u?u?v?v

dx?dy???dv?dx?dy??x?y?x?y

?????????

FxFFdydyd2y??

???F(x,y)?0??2?(?x)?(?x)?

dxFy?xFy?yFydxdxFyFx?z?z

???F(x,y,z)?0????

?xFz?yFz

?????????????

??????

?????

1(tgx)??sec2x(arcsinx)??

(ctgx)???csc2x?x2

(secx)??secx?tgx(arccosx)???

1

(cscx)???cscx?ctgx?x2(ax)??axlna

(arctgx)??

1

1?x2

(logx)??

1

axlna

(arcctgx)???

1

1?x??????

?tgxdx??lncosx?C?ctgxdx?lnsinx?C

?dxcos2x??sec2

xdx?tgx?C?dx?secxdx?lnsecx??Csin2x??csc2

xdx??ctgx?C

?cscxdx?lncscx?ctgx?C?secx?tgxdx?secx?C

?dx1?cscx?ctgxdx??cscx?C

a2

?x2

?aarctgx

a

?C??ax

dx1xdx?ax

lna?Cx?a??a

2alnx?a?C?shxdx?chx?C

?dx1a?a2?x2?x

2alna?x?C?chxdx?shx?C

?dxa2?x2

?arcsinx

a

?C?dx

x2

?a2

?ln(x?

x2?a2)?C

??2

n

2

In??sinxdx??cosnxdx?

n?1

nIn?2

?x2

?a2

dx?xx2?a2?a2

ln(x?x2?a222)?C

?x2?a2

dx?x22a22x?a?2lnx?x2?a2?C

?

a2?x2

dx?x22a2x2a?x?2arcsina

?C

???????????

sinx?2u1?u2x2du

1?u2??cosx?1?u2??u?tg2??dx?1?u2

??????????????

:shx?

ex?e?x

????2lim

sinx

x?0x

?1

?

ex?e?x

????:chx2

lim x??(1?1

x

)x?e?2.718281828459045... ????:thx?shxex?e?x

chx?

ex?e?x

arshx?ln(x?x2?1? archx??ln(x?x2?1)

arthx?11?x 2ln

1?x

??????? �?????

�?????? �???????

sin(???)?sin?cos??cos?sin?sin??sin??2sin

???

cos(???)?cos?cos??sin?sin?

2cos

???

2tg(tg??tg?

sin??sin??2cos??????

???)?

1?tg??tg?2sin

2cos??cos??2cos??????

ctg(???)?

ctg??ctg??1

2cos

2ctg??ctg?

cos??cos??2sin??????

2sin

2

�?????

sin2??2sin?cos?

cos2??2cos2??1?1?2sin2??cos2??sin2?sin3??3sin??4sin3?ctg2??

ctg2??1

cos3??4cos3??3cos?2ctg??

3tg??tg3tg3?tg2??

2tg?

1?3tg2?

1?tg?

�?????

sin?

?2

??2????????????cos2??2

tg?2??1?cos??1?cos?sin??sin?1?cos???ctg?1?cos?sin?2??1?cos??sin??1?cos?

�?????

asinA?bsinB?csinC?2R �?????c2?a2?b2?2abcosC

�????????arcsinx?

?

2

?arccosx???arctgx?

?

2

?arcctgx

??????��?????Leibniz????

(uv)

(n)

??n

Ck(n?k)(k)

nuvk?0

?u(n)v?nu(n?1)v??

n(n?1)u(n?2)v?????n(n?1)?(n?k?1)k!

u(n?k)v(k)

???uv(n)

2!??????????

?????????f(b)?f(a)?f?(?)(b?a)f(b)?f(a)f?(?F(b)?F(a)?

)

F?(?)

?F(x)?x???????????????????

???

??????ds??y?2dx,??y??tg??

??

?s

.??:?M??M???????????????s?MM????M?????K??lim??s?0?s?d?

ds?y??1?y?2)

3.

???K?0;???a???K?1

a

.

?????????

b

?????f(x)?

b?a

n

(y0?y1???yn?1)a

b

?????f(x)?

b?aa

n[1

2(y0?yn

)?y1???yn?1]

b

??????f(x)?

b?a

a

3n

[(y0?yn)?2(y2?y4???yn?2)?4(y1?y3???yn?1)]??????????

??W?F?s????F?p?A

???F?km1m

2r

2,k?????

?1b

b?a?f(x)dx

a1bb?a?f2

(t)dt

a

????????????

??2?????d?M1M2?(x2?x1)2?(y2?y221)?(z2?z1)?????????Prju?cos?,?u?????Prj??a???u(a12)?Prja1?Prjaa??b??a??b?

2cos??axbx?ayby?azbz,?????,?????????cos??

axbx?ayby?azbz

a22x?a2y?a2z?b2x?by?b2

zc??a??b?ijk

?aa??????xayz,c?a?bsin?.??????v?w?r.

bxbybz

axa???????[a?b?c?]?(a??b?)?c?yaz

?b???

xbybz?a?b?ccos?,??????

cxcycz ???????????

??????

1?????A(x?xy?yn?

0)?B(0)?C(z?z0)?0????{A,B,C},M0(x0,y0,z0)2??????Ax?By?Cz?D?0

3xa?yz

b?c

?1

???????????????d?

Ax0?By0?Cz0?D

A2?B2?C2

x?x0m?y?y?x?x0?mt

0n?z?z0p?t,???s?{m,n,p};??????

?y?y0?nt

??

z?z0?pt

?????

1x2a?y2z2

b?c?12x22p?y2

2q

?z?,p,q???

3?????

x2a2?y2b2?z2

c2?1

x2y2z2

a2

?b2?c

2??????1

??????????

????dz?

?z?xdx??z?ydy???du??u?xdx??u?ydy??u?z

dz??????????z?dz?fx(x,y)?x?fy(x,y)?y???????????

z?f[u(t),v(t)]dz?z?u?z?v

dt??u??t??v??t

?

z?f[u(x,y),v(x,y)]?z?z?u?z?v

?x??u??x??v?

?x

?u?u(x,y)?v?v(x,y)??

du??u?xdx??u?ydy???dv??v?xdx??v

?ydy?

?????????

???F(x,y)?0dyFxd2y?Fx?Fdy

dx??F2?(?)(?x)?

ydx?xFy?yFydx???F(x,y,z)?0?zFF?x??x?z

F??y

z?yFz


?????????????????
?????http://www.850500.com/news/5546.html
?????,??!