x2 dla 0 d" x d" 1
2 e Ą
f(x)dx f(x) = | ln x|dx x�sgn(cos x)dx
1
0 0
2 - x dla 1 < x d" 2 2
Ą
3 1 2 4
e2x dx
2
"
sgn(x - x3)dx (x + 1) cos xdx xe-xdx dx
0 0 0 0 1+ex 0 1+ x
Ą
" Ą
ln 2 9
(1-x)2
cos x
2
2
"
ex - 1dx Ą dx sin2 x cos xdx dx
0 25+sin2 x 0 4 x x
6
"
"
"
n
n
n
1
limn" 14+24+...+n4 = limn" e+ e2+...+ en = e - 1
n5 5 n
4 4
f(x)dx = 4 g(x)dx = 2
-1 -1
4 4
f" f(x)-g(x)
(x)
dx dx
-1 -1 2
2
y = f(x) [-a, a]
a a
f(x)dx = 2 f(x)dx
-a 0
y = f(x) [-a, a]
a
f(x)dx = 0
-a
f [a, b]
b 1
f(x)dx = (b - a) f(a + (b - a)x)dx.
a 0
40Ą Ą
| sin x|dx = 50 | sin x|dx
-10Ą 0
1 dla x " N
/
100
g(x)dx g(x) =
0
2 dla x " N
x dla x " Z \ {0}
/
10
g(x)dx g(x) =
1
-2
dla x " Z \ {0}
x
2
2 2 1 1
dx dx
"
e-x sin xdx e-x sin xdx
1 0 0
1+x2 1 x
ńł
1 dla x " [-1, 0]
�ł
x
F (x) = f(t)dt x " [-1, 2] f(x) = x dla x " (0, 1]
-1
ół
x2 dla x " (1, 2]
ńł
x
�ł - 1 dla x " [0, 1]
x
F (x) = f(t)dt x " [0, 3] f(x) = -2x + 4 dla x " (1, 2]
0
ół
1 dla x " (2, 3]
x
f(x) = (|t - 1| + |t + 1|)dt x e" 0
0
x
g(x) : [0, +") R g(x) = ||t - 1| - 2|dt
0
(5, 1) x(t) = 2 + 5 cos t y(t) = -3 + 5 sin t
"
(2, 3) x(t) = 2 cos t y(t) = 2 sin t
x(t) = 3t y(t) = 6t - t2 x(t) = cos t y(t) = sin 2t x(t) = t3 + 1 y(t) = t2
x(t) = t - sin t y(t) = 1 - cos t
t
x(t) = 3(2 cos t - cos2t) y(t) = 3(2 sin t - sin 2t) P = (-9, 0)
x(t) = t2 + 2t y(t) = t3 + t P = (3, 2)
x(t) = 2tgt y(t) = 2 sin2 t + sin 2t P = (2, 2)
y2
1 x2
x = -1 x = 1 y = y = sin x x " [0, 2Ą] Ox + = 1
1+x2 9 4
y = x3 + x2 - 2x x " [-2, 2] y2 = x x = 8
y = sin x y = cos (2x) y = x2 y = 2x2 y = 8 x e" 0
x x2 x2
y = x2 - 1 y = 5 - x y = - 1 y = , y = 3x - (x - 1)2 + (y - 2)2 = 4
5 4 2
y = -x2+4x-3 y = 6-x y = 4x-3 xy = 4, x+y = 5 y2 = 2x, x2+y2-4x = 0
9y2 = x(x - 3)2 y2(2 - x) = x(1 - x)2
5
f = const f(x)dx = 0

-5
"
4
y = x2 x " [0, 2] y2 = x2 x = y = 2x - x2 x " [0, 1]
3
" " "
"
Ą
y = ln (sin x) x " [Ą , ] y = ln x x " [ 3, 2 2] y = x - x2 + arcsin x
3 2
ex+1
y = ln
ex-1
y = 1-x2 K = {(x, y) : -2 d" x d" 2, -2 d"
y d" 2}
Ox y2 = 4x 0 d" x d" 3
y = sin x x = 0 x = Ą
y = x2, y2 = x
a d" x d" b 0 d" y d" e-x
1
y = x " [2, 4]
x-1
"
y = e-x sin x x " [0, Ą]
" "
3
y = x, y = x
Ł
y = x3 0 d" x d" 1
x2 + (y - 3)2 = 4
x2 3
y = y =
2 2
y2
x2 x2 z2
+ = 1, + = 1
a2 b2 a2 b2
x = 0 y2 = 4-x
x2
y = a - y = a - x
a
x2 + y2 + z2 = R2
y2
x2
+ = 1
a2 b2
16x2 + 8y2 = 144
(x0, y0) a, b
(x - x0)2 (y - y0)2
+ = 1
a2 b2
x = a(t - sin t) y = a(1 - cos t)
t " [0, 2Ą]
"
1
x = t2, y = t - t3 t " [0, 3]
3
x(t) = a cos3 t y(t) = a sin3 t
x(t) = r cos t y(t) = r sin t r
x = r(cos t + t sin t), y = r(sin t - t cos t), t " [0, Ą], r > 0
"
t3
x = t2, y = t - t " [0, 3]
3
t 2
x = cos t + ln(tg2), y = sin t t " [Ą , Ą]
2 3
x = R cos t y = R sin t t " [0, Ą]
x = a(t - sin t) y = a(1 - cos t) t " [0, 2Ą] a > 0
x = a cos3 t y = a sin3 t
 r = a(1 + cos �) a > 0 � " [0, 2Ą]
r = a sin 2� a > 0 � " [0, 2Ą]
Ą
r = a(1 + cos �) a > 0, t " [0, ]
2
(x2 + y2)2 = a2(x2 - y2) a > 0
(x2 + y2)3 = 4a2xy(x2 - y2) a > 0
+" +" 0 +" +" +"
dx dx dx ln x
"
sin xdx dx x sin xdx
1 x4 0 3x+1 -" 1 x 1 x 0
+" +" " -1 +"
dx xdx dx
"
e-x sin xdx e-2xdx
3
0 -" x2+2x+2 0 x2+4 -"
3x-5 -"
1
1
+" +" - +" +"
x
dx e dx
2
"
x ln (x2 + 1)dx arcctgxdx dx
-" 0 -" x2+x+1 1 x2 1
x 1+x5+x10
+"
dx
2 x2+x-2
x = 1
1
Ox 1 d" x < " y = [1, +")
x2(x+1)
y = 0
x
y =
x4+1
3
2 e 0 1 1 1 Ą
dx dx x-1 dx dx dx
2
" " " "
dx ln xdx
3
1 (x-1)2 1 0 Ą sin2 x
4x-4x2 -1
1-x2 0
x ln x -1
x5
3 1
Ą 1 1 16
dx
"cos xdx Ą 1 1 e dx dx x "
sin dx dx
3 3
0 0 x2 x 0 x ln2 x 0 x2-4x+3 0 1-x 0
1-2 sin x
x4
1 2 2
dx dx dx
" " "
1
1 1 -x2+3x-2
1-x2 arcsin x x2-1
2
Ą
dx
2
t = tgx
0 a2 cos2 x+b2 sin2 x
 x = 5-1
t
5
dx
"
0
25-x2
Ox 0 d" x d" 9
1
"
x = 0 x = 9 y =
3
x-1


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