N. PISKUNOV
DIFFERENTIAL
INTEGRAL
CALCULUS
M—y\
(1)
2" > T’ nlog2>logTor n>T^2’
_I2_ > _J > _L2_
|a(*)|
(1+^)"
(!+--)
=,l™ (^r-^.o+Tj'
-,az('+r)'('+T)^-'-e
'.?.('+7);K)*('4r
ln*=i,og*
/(**)
a, P, y, ...
Um = lim ( ! _I) = !-Hç, 1 = j _ j = °
(2)
«' u" & y * yxf fa
(1)
^ = /'(*o) + Y .
A y = nxtt~l Ax+ /l(|t~l) xn~2 (A*)* + ... + ( Ax)n
-jr~sin (*+¥) (*+t)
=L^)v+u Vm^+litn Au lim^
2 VT
S.-L* (,+Ü)
(1)
a (wTTv ,nu)
/7=T.. WCget
*=
(*vi)
V i —*2
but
y =—■=f
y i—x*
,.,M, I ».-/.<«>*.<*>
(i)
ÿ = [<£(*)] (2)
0 = ^(0 I
16=4-
(3').
rw-î
y=f(x)
y"=(y')' = /"(x)
r(x).
—sin B( 277 )* cos “ (—&7r){dx)t
nir m ' "
n ” /',<*.)
N = V y\+(y^Y = \yiV\+y?\
iV
x(tan*+-dh)-
3 v (*—1)2 v
TT+ï
V^+b*a+±, l/JTb*a+±s
m^i=r,c)
'■(4)
(1)
(2)
(3)
w .
(7)
m-«) ('-•) < £$—fê- < (■4+e> c+e>
rw
(8)
+ ••• + ^ r(a) + R„(X) (6)
(7)
/•■' (o=- r (o+r (0 -V f (o+2-VL f w
f'"( <)+••• + fin) w + r (o
S*zzîïLfin+»(t) + ln+^X)Xty,Q
F' (0 = - —r2 /(n+1, (0 + (±=r1 Q
(8)
R* (*)=>j),)r/(n+1) [«+0 <*-«)]
+i£^r(a) + i£^,..*..[a+e(,-a)] (9)
+Çfin) (°)+(5nir ^ (0*) 0°)
/(*)=«*. /( 0) = 1
*=1 + I+i+'5ï+" •+¥
(1)
151<1*1
+o, - y~2
(1)
(2)
/(x, +AxX/fo)
/(Xj + Ax)—/(xx) < 0
o
/'W
n* i)=o
4KI
6 X
te')*<0>°> (y\>o<0
/'(a) = f(a) = ...=r(a) = 0
f(x) = f(a) + ix-V*“f«+»(l) (2)
f(x)-f(a) = ^=^r+>>( |) (2')
y—ÿ=f(x)—f(x0)—f (xt) (x—x0)
\n
r2
{,=(*—u3
*-+•L * * J
(4)
(5)
—;—J
[-il-
K?-—■
(i)
V = f(x)
(2)
(3)
'(0q>'(0—q>" (01'(<)
/ï
= î/-.
' 2(/,
mnÈ y-i+fif)’
*- /P)
K _ l*q>| _|A(pf
(3)
(1)
de* de* " de ïi¥ = S- sin0+2 tbt cos 0—p sin 0
(4)
^=a, 4£=0
(O
9 = f(x) (3)
•fl, P = 0=F
/1+/1
(T)
dp
(£)•-(£)■
/<*) = o
1 *i—*! I = V (ûi—a,)2 + (f»i—b2)2
(6)
/(«i) = 0. /' (ax) — 0, f"(a1) = 0, .... /**■-« (fl|) = 0
... (x—a,)*'(x2 + pjx+ ?!)'* ... (x2 + psx + çs)‘s
*/o =
z = f(x, y)
*=/(*. y)
f(xo, y0, y, ...)
(2)
+p‘
d*=éAx+éAy
(1)
(2) (3)
(4)
T
|Ay|+ • •• +
(1)
(2)
V (75)*—(32)* 75 Y(75)* — (32)*
|ô*x|
(3)
|A*/|
(4)
|ô*u|
I A*y| \y\
6**1 +|«VI
|ô*«|=16*x|+|à*y|
Â7 = dï--ÂF + *T ÂF+^-sj 4-Y*-5J-
(4)
(4')
d±=2x
(6)
(2)
f'x(x, y + Ay)—f'x(x, y) = Ayf’xy (x, ÿ)
A = AxAyf’xy{x, ÿ) (1)
A=[f(x+Ax, y+Ay)—f(x, y+Ay)] — [f(x+Ax, y)—f(x, y)]
♦ (ÿ) = /(* + Ax, y)—f(x, y)
A=ÿ(y + Ay) —ÿ(y)
V (ÿ) = fy{x + Ax, y)—fy(x, y)
f'y(x + Ax, y) — fy(x, y) = Axf’yx(x, ÿ)
A = Ay Axf’yx (x, y) (2)
Ax Ayfxu (x, ÿ) = Ay Axfyx (x, ÿ)
^4+1+9 vu
vi4 /t?
(-t)»-* (§)*=2
f2.
+2
^-27T
y)
f (*. y) =
+ (y—b)2 flv (a, &)] +^- [(x—a)3f'^x(li, b)
/(*o. y0)>f(x, y)
/(*«,. &>)
{t) i, tHen theremv
/(*.+A*, y.+Aÿ)=I (*.. y,)+dl y"1 A* + * ,l) Ai, +
=A*a+2 ^ A* Ay+ |f® Ay2 ] + a»(Ap)î
(1)
A/=|(Ap)® cos V+B sin 9 + 2ct(>Ap] {3)
a a .. (a a\
(6)
F(x„ x xn, Xlt .... K) = f(*i + ••
x„:
(9)
ÊL
+XZ+
^[i-g <«/+*)] =°
» [|-â(x+z)]=°
i(y+*)=1’ ï(*+*)=l- i
(4)
S [yt—a. b,c, ...)] d
Z
(5)
S (a. b) = 2 lw-(«/ + »)]*
23*/“H. i]y/=io
(§).„-<* - (SU-
£+üL_i
r =
Ar = r(/ + A/)—r(t)
+ [♦(/+ao-♦(<)] y
+ [x(f + A<) — x(0]k
%\-WW+fe' (0]2+[x' (O]2
«M*. 0. 2) = 0, .,(*, y, z) = 0 (6)
(12)
¥*-o.
—sr~=Ma+
=(M+M+ %ik) (M+M+ Xa*)+(M+M+x,*) (M+M'+x^)
As Tb
(2)
o.
(4)
(5)
(6)
(7)
(8)
I5-I- Bm |£
0.
(§)’
R,= {($)7
fdr
mi
{[•P' MP+N»' (0P}V*
(12)
- /(IHIHI7
(i)
(2)
£x»=-i-ffx» = 0
(3)
LL
x
(2)
l»l-/(ï),+(S),+(ï)Vo
z-z=TxAx+%*y
(7)
(f+c)-,.
«P (*)—
2. jf=ln|*| + C.
10. ^axdx = -^ + C.
J yT^*5
lTfe-lnU+KlrïS!|+c
(S IM*)+ /•(*)]<**) =fi (*) + /* (*)
(5)
4-«
($ / [q> (0]
=/[
=^+c=|5I,,,+c.
/,
-S;
l-\-bx-\-c J ax2 + bx + c
_ A C 2ax+b d (B Ab\ P dx
2 2a J ax2 + bx-l-ca 2a J J ax2-\-bx+c
IS£SFp-If-tal'l+C = l"l“‘+^+^+C
J_ Ç(2x—2)dx f» dx
-f
y xa-t-4x+ io
■î
f x2dX = f* xdX - = —* Va2—x2 + f Va2—x2dx J Va2-x2 J \fa2-x2
2 J (x* + px+q)" T\ 2 J J (x* + px+q)*
fC
= i[.
(<2 + m2)*-1
= J_ f f d(<2+m2)
O)
r
-_L_(W—!—)
C—* 1
+±_L_r—? f_ÜL_1
s
H
(5)
-4ini*+' i+4,nijc—2i+c
ç * r** f- •
J»'** “J & J,="
fQ- y 1+*+**)* ^
J x* y i +*+**
l+*+*2=*2/2 + 2;irf + l, * = àx=~^à.-£-dt
i—v'r+*+*î=■
J *2 Vl+x+* J (!-/•)• (»-!)• (<*-< +1)(1-/*)2
=+2I
J j/> + 3*_4
'-/Si
+c
cos T+Sin T
S 2^*=! ï+t 7+î *-1 (‘-2+rjh) *
I Si ifïî-f '■ "+'*>"'-T+f+c
ax2 + bx + c = a^x-+(c—
x+£ = t’ dx = dt
fis.. te. I,*v~+C. s. j(Js + -jLr + 2)*.
+ c.
.. w-m
J K^l + COS2^
J V(l-x2)4
i££ÜE£_ , i lB|i-*|,r Kî^i+Tln|T+3i|+c-
,7,‘ I ^eî/fx~dx'Ans' Z*^x'~i^xis +c-172• j*
V*
yr+x— y 1—x
yr+^+yr—;
i4ns. 14 x — ÿ V *+y 'V* - y V ** +ÿ k'V] +C.
2 tan-J
*0 < *1 < *2 < " • • < X„
(1)
(2)
(5)
sa = f(ll)Axl + f(ti)Ax2 + ...+f(ln)Axn = £f(ti)àxi (1)
Sn = 2 Axf
(9)
(10)
s" = * Lna + -4 H —Lû+"^ 2 J
2 î (6,) A*,. = 2 / «,) A*,. + 2 / (i,) Ax,
/«)
$/ [9(0]
= F MP)]-F Ma)]
one says that J f(x)dx does not exist or diverges.
ÎT
b
!
f î±l«
J VI?
M(X„ i/o). i/i). i/o)
j f (x) dx « — (yt + 4y3 + y*)
«-if
(1)
(2)
(3)
J [f'a (X, a)4-e] dx = J fa(x, a)dx + ^edx
[i
<3>
^r = wSf(x’ a)dx = --fc$ï(x> o)dx = — f[a(a),a]
£ = /(*) = /[
P = /(6)
Qn = ÿ H p? Ae,.=-1X If (ë))]2 A0,
jfpMe
' Q = i-jp’d0 (1)
Q=tJ[/( °)lsd0 0')
sR=i/i+lrWAx,.
*=ç(0. y='H0. z = X(0 (6)
(•£)' ■+ (%)' - \r w+[f (»)]*=p'*+p*
Q =
Q(S/)
Q (h) A.v,-
v„ = S Q (h) Ax,
Ay,- f(xt)—f(xj-1) ; ,, .
/>„=K s KT+TWax,. (1)
(3)
VTp
F (h) As,
AAk^dr=-ke'e*T\r,rrke'e*(^--k)
(i)
(2) 
