Download Integrals of simple functions

Integrals of simple functions
1
Rational functions
Z
(∗)
dx =
Z
xn dx =
(∗)
Z
1
dx =
x
Z
du
=
2
a + u2
(∗)
(∗)
2
x+C
xn+1
+C
n+1
ln |x| + C
1
u
arctan + C
a
a
Logarithms
Z
ln x dx
=
x ln x − x + C
logb x dx
=
x logb x − x logb e + C
Z
3
Exponential functions
Z
(∗)
ex dx =
ex + C
ax dx =
ax
+C
ln a
Z
if n 6= −1
4
Irrational functions
Z
du
− u2
Z
−du
√
a2 − u2
Z
du
√
u u2 − a2
√
(∗)
5
a2
u
+C
a
u
= arccos + C
a
|u|
1
arcsec
+C
=
a
a
= arcsin
Trigonometric functions
Z
(∗)
sin x dx =
− cos x + C
cos x dx =
sin x + C
tan x dx =
− ln |cos x| + C
cot x dx =
ln |sin x| + C
sec x dx =
ln |sec x + tan x| + C
csc x dx =
− ln |csc x + cot x| + C
Z
(∗)
Z
Z
Z
Z
Z
sec2 x dx =
tan x + C
csc2 x dx =
− cot x + C
Z
Z
sin2 x dx =
Z
cos2 x dx =
Z
sinn x dx =
Z
cosn x dx =
Z
arctan x dx =
1
(x − sin x cos x) + C
2
1
(x + sin x cos x) + C
2
Z
sinn−1 x cos x n − 1
−
+
sinn−2 x dx
n
n
Z
cosn−1 x sin x n − 1
−
+
cosn−2 x dx
n
n
p
x arctan x + ln 1 + x2 + C
Note. The most important integrals are indicated with (∗)