"
"
"
"
"
 F f (a, b)

F (x) = f(x)
x " (a, b)
(a, b)
F f C " R
f

f(x)dx,
{F (x) + C} .

f(x)dx = F (x) + C �!�! [F (x) + C] = f(x).

0dx = C C " R

xa+1
xadx = + C x > 0 a = 0

a + 1

a = 0 dx = x + C

a = 1 xdx = x2 + C

"
1 dx
a = - " = 2 x + C x > 0
2 x

dx 1
a = -2 = - + C x = 0

x2 x

sin xdx = - cos x + C

cos xdx = sin x + C

1
dx = x + C cos x = 0

cos2 x

1
dx = - x + C sin x = 0

sin2 x

1
" dx = arcsin x + C -1 < x < 1
1 - x2

1
dx = x + C
1 + x2

exdx = ex + C

ax
axdx = a > 0 a = 1

ln a

1
dx = ln |x| + C x = 0

x

"
dx
" = ln (x + x2 + k) + C
x2 + k

x5dx

"
3
xdx

dx
x4

exdx

dx
ex
f g
(f(x) + g(x))dx = f(x)dx + (g(x)dx

(f(x) - g(x))dx = f(x)dx - (g(x)dx

c � f(x)dx = c � f(x)dx c " R

(x - 2ex)dx

x2 - x + 1
" dx
x

x2
dx
1 + x2
f : (a, b) R (a, b)
� : (A, B) (a, b) (A, B)

f(x)dx = f(�(t))� (t)dt = F (�(t)) + C,
F f C " R

x7dx
"
1 - x16

cos7 xdx

"
x 1 + xdx
"
6
xdx
"
3
1 + x

dx
1 + e3x
f g

f(x)g (x)dx = f(x)g(x) - f (x)g(x)dx.

x sin xdx

x2e-xdx

xdx
cos2x

x arctan xdx

W1(x) anxn + an-1xn-1 + � � � + a1x + a0
dx = dx.
W2(x) bmxm + bm-1xm-1 + � � � + b1x + b0
 n m
(n < m)
n < m
A Bx + C
,
(ax + b)k (cx2 + dx + e)p
A, B, C, a, b, c, d, e " R " = d2 - 4ce < 0
cx2 + dx + e k, p " N
f

f (x)dx
= ln |f(x)| + C.
f(x)

x2 + 2
dx
x + 2

x2 - 4
dx
x - 1

x3
dx
x2 - 3x + 2

x5 + x4 - 8
dx
x3 - 4x

x4 + 6x3 + 10x2 + x
dx
x2 + 6x + 10

3x + 1
dx
x2 + 4x + 4

6x3 + 4x + 1
dx
x4 + x2
x
m
m, n " N
n
x = tN,
m
N
n

dx
" " .
3
x + x
x
ax + b
ax + b
, ad - bc = 0,

cx + d
m/n m, n " N
ax + b = tN,
ax + b
= tN,
cx + d
N m/n
"
4
3x - 7dx

dx
" dx
3
4 - 5x

"
x 2x - 10dx


x + 1 dx
3
�
x - 1 x + 1
" x
ax2 + bx + c


dx
x "
" = ln + x2 + k + C

x2 + k

dx x
" = arcsin + C.
|a|
a2 - x2
f


f (x)dx
= 2 f(x) + C.
f(x)

dx
"
x2 - 6x + 15

dx
"
4 - 2x - x2

(3x + 1)dx
"
x2 + 5x - 10

Wn(x)
" dx,
ax2 + bx + c
Wn(x) n

"
Wn(x) dx
" dx = Wn-1(x) ax2 + bx + c + A " ,
ax2 + bx + c ax2 + bx + c
Wn-1(x) n - 1 A

6x3 - 22x2 + 21x - 7
" dx
x2 - 4x + 3

"
(3x - 2) x2 - 2xdx
x
tan = t
2

R(sin x, cos x, tan x)dx,
R(u, v, w) u, v
w
x 2dt
tan = t, x = 2 arctan t dx = .
2 1 + t2
2t 1 - t2 2t
sin x = , cos x = tan x = .
1 + t2 1 + t2 1 - t2

dx
2 + cos x

2 + sin x
sin x(1 + cos x)

R(sin2 x, cos2 x, sin x cos x)dx,
R(u, v, w) u, v
w
dt
tan x = t, x = arctan t dx = .
1 + t2
t2 t 1
sin2 x = , sin x cos x = cos2 x = .
1 + t2 1 + t2 1 + t2

dx
1 + 2 cos2 x

1 + sin x cos x
dx
(2 + cos2 x)(1 + sin2 x)


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