I0
b
f f(x)dx
a
P y = f(x), Ox x = a x = b
b
P = f(x)dx, f(x) e" 0 x " [a, b].
a
f [a, b] P
b
y = f(x), Ox x = a x = b - f(x)dx
a
b
P = - f(x)dx, f(x) d" 0 x " [a, b].
a
f [a, b] P
y = f(x), Ox x = a x = b
Ox
c d b
P = P1 + P2 + P3 = f(x)dx - f(x)dx + f(x)dx.
a c d
y = f(x) y = g(x) x " [a, b] f(x) e" g(x)
f g,
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x = a, x = b
b
P = [f(x) - g(x)]dx.
a
f g [a, b], k = const. c " [a, b].
b a
f(x)dx = - f(x)dx;
a b
b b
k · f(x)dx = k f(x)dx;
a a
b b b
f(x) + g(x) dx = f(x)dx + g(x)dx;
a a a
b c b
f(x)dx = f(x)dx + f(x)dx.
a a c
f(x) [a, b],
x
F (x) = f(t)dt, x " [a, b]
a
[a, b]
f [a, b] F f,
b
f(x)dx = F (b) - F (a).
a
f(x)
x = Ć(t), t " [Ä…, ²],
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" Ć(t) " [a, b] t " [Ä…, ²];
" Ć(Ä…) = a, Ć(²) = b;
" Ć (t) [Ä…, ²]
b ²
f(x)dx = f Ć(t) Ć (t)dt.
a Ä…
r = f(É), É " [Ä…, ²] f(É)
[Ä…, ²] (Ä…, ²), r = f(É)
f(Ä…) r(²)
²
P = f2(É)dÉ.
Ä…
“ : y = f(x) x " [a, b]
b
|“| = 1 + (f (x))2dx.
a
r = f(É) Ä… d" É d" ²
²
2
dr
|“| = r2 + dÉ
dÉ
Ä…
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V
Ox N : a d" x d" b, 0 d" y d" f(x)
b
V = Ä„ f2(x)dx,
a
Oy N : 0 d" a d" x d" b, 0 d" y d" f(x)
y = f(x)
b
V = 2Ä„ xf(x)dx.
a
f(x) Ox Oy
Ox f(x) a d" x d" b
b
P = 2Ä„ f(x) 1 + (f (x))2dx,
a
Oy f(x) 0 d" a d" x d" b
b
P = 2Ä„ x 1 + (f (x))2dx.
a
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f(x) Ox Oy
f [a, ").
f [a, ")
" B
f(x)dx := lim f(x)dx.
B"
a a
(-", b] :
b b
f(x)dx := lim f(x)dx.
A-"
-" A
f (a, b]
a.
f (a, b]
b b
f(x)dx := lim f(x)dx.
ta+
a t
f [a, b) b
b :
b t
f(x)dx := lim f(x)dx.
tb-
a a
c [a, b]
b t b
f(x)dx := lim f(x)dx + lim f(x)dx.
tc- tc+
a a
c+
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4 1 3
dx x-1 1
a) ; b) dx; c) dx;
x2+3x+2 x+1 x2+9
3 0 0
Ä„/3 Ä„/2
6
1+cos2 x 6x
"
e) dx; f) dx; g) sin3 x cos xdx;
3
1+cos 2x
(x2+4)5
0 0
Ä„/6
e 4 Ä„
dx
"
h) ln xdx; i) ; j) sin3 xdx;
1+ 2x+1
0 0
1/e
2 3 0
"
x
"
k) 4 - x2dx, (t = 2 sin x); h) dx; l) xe-xdx;
1+x
0 1 -1
Ä„/3
2 2
1+tg2 x
x2+1
"
m) x ln xdx; n) dx; o) dx.
3
(1+tg x)2
x3+3x+1
1 1
Ä„/4
D
y = ln x, x = e, y = 0; y = x2 - 6x + 5, y = 5 - x;
y = x2, y = 2x2, y = 8, x e" 0; y = x3 - x2 - x, y = x;
y = x3, y = 4x2 - 3x; y2 = 4 + x, y2 + x = 2;
y = x2 - x - 6, y = -x2 + 5x + 14; y2 = 2x, x = 8;
1
y = cos x, y = x2 + 2; y = 1 - x; x2 + y2 = 4x, y2 = 2x.
4
r = f(É)
r(Ä…) r(²) :
"
Ä„ Ä„
f(É) = 3 - cos 2É, Ä… = 0, ² = ; f(É) = 2 cos2 É, Ä… = 0, ² = ;
2 4
a Ä„ Ä„ 2
f(É) = , a = const. Ä… = , ² = ; f(É) = , Ä… = 0, ² = 2Ä„.
É 4 2 2+cos É
" 1
Ä„
y = 1 - x2 0 d" x d" ; y = ln(cos x) 0 d" x d" ;
2 3
2 3
1
3 2
y = (ex + e-x) 0 d" x d" 1; y = (4 - x ) 1 d" x d" 8.
2
1 1
f(É) = , Ä… = , ² = 2; f(É) = É, Ä… = 0, ² = 2Ä„;
É 2
f(É) = 2 - 2 sin É, Ä… = 0, ² = 2Ä„; f(É) = a sin3 É , Ä… = 0, ² = 3Ä„;
3
N OX, N
y = 2x - x2, y = 0; y = x2 - 4x, y = 0;
"
y = sin2 x, x = 0, x = Ä„; y = x2, y = x.
N Oy :
Ä„
N : 0 d" y d" tg x2, 0 d" x d" N : y2 = 4 - x, x = 0.
3
R;
x2 + y2 + z2 = R2 x2 = y2 + z2, x e" 0.
"
y = ln x 1 d" x d" 3 Oy; y = 6x 0 d" x d" 1 Ox;
"
y = sin x 0 d" x d" Ä„ Ox; y = 25 - x2 - 2 d" x d"
3 Ox.
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R; r h.
" " 0
1
a) e-2xdx; b) dx; c) (x - 2)e3x+!dx;
x
0 1 -"
" " "
x 1
d) dx; e) dx; f) x cos2 xdx;
x2+4 x2+9
"
0 -"
Ä„
" " "
1 dx
"dx
g) dx; h) dx; i) dx.
x2(x+1) x2+6x+12
x 1+x2
1 -" 1
-1/2
4 1
1 ln x 1
" "
a) dx; b) dx; c) dx;
x x x 2x+1
0 0 0
3Ä„/2
3 3
dx x 1
"
d) ; e) dx; f) dx;
x2-1 sin2 x
9-x2
-3 0 Ä„
3 3 1
1 dx 1
"
g) dx; h) ; i) dx.
3
x(x-3) x ln x
(x-2)2
2 1 0
"
3
7
t t
2
2t + 40 t,
t0 = 0 t
2
3
20 - 5t
3
4
40 - t t