xn+1
xn dx= — n+1 (n —1)
dx = x
x
S
sin x dx= — cos x
cos x dx= sin x
tan x dx= — in cos x
S
cot x dx = In sin x
. c dx — tan x
cos x
C dx — cot x
j sin' x
ex dx= ex
dx
J 1+0
=arctanx
arcsin x
In (x+ 17 Z2 -
V x2-1
Integration by parts:
u du = uv— v du
9. Derivatives and Integrals
Function
Derivative
Function
Derivative
Function
Derivative
xn
1
x
1
xn
1 7x
ex
enx
ax
In x
nxn-1
1
-- T
x
n
_
sin x
cos x
tan x
cot x
cos x
—sin x
arcsin x
arccos x
aretan x
1
V 1— x2
1
cos
1
2 x
1
1 / 1— x 2
1
xn+i
1
sing x
1-{-x'
- . f
2 v x
ex
nenx
ax Ina
1
—
x
Vu
In u
u
_
v
u'
arccot x
sinh x
cosh x
tank x
cosh x
2 Va
+
u'
— a
vu'v'u
1 x2
cosh x
sinh x
1
cosha x
1
v2
sinh2 x
Some Definite Integrals
1
1, n=0
et e-x dx= til
i ll ii-, n= 1/2
1, n=1
o
2, n=2
00
I 1/2. 1iTT, u= 0
5 x n tr* xa dx = i "2' -: n = 1
1 /417 a., n=2
o
1/2,
n=3
0°
c zn dx —
2.31, n=1/2
n2/6, n=1
2.405, n-= 2
n4/15, n= 3
24.9, n=4
c&
r x3dx
0.225, a= 1
1.18, a= 2
2.56, cc= 3
4.91, a= 5
6.43, cc= 10
j ex— 1
o
ex
1
o