Integral Formulas
Z
Z
1.
u dv = uv − v du
Z
un+1
2.
un du =
+ C (n ̸= 1)
n+1
Z
du
3.
= ln |u| + C
Z u
eu du = eu + C
4.
Z
5.
au du =
au
+C
ln a
Z
8.
9.
cos u du = sin u + C
csc2 u du = − cot u + C
Z
10.
sec u tan u du = sec u + C
Z
csc u cot u du = − csc u + C
11.
Z
tan u du = ln | sec u| + C
12.
Z
cot u du = ln | sin u| + C
13.
Z
14.
Z
Z
2
sec u du = tan u + C
Z
sin u du = − cos u + C
6.
7.
Z
sec u du = ln | sec u + tan u| +
C
csc u du = ln | csc u−cot u|+C
15.
Z
u
du
+C
= sin−1
a
a2 − u2
Z
du
1
−1 u
17.
=
+C
tan
a2 + u2
a
a
Z
u
1
du
√
18.
= sec−1
+
a
a
u u2 − a2
C
Z
du
1
u+a
19.
=
+C
ln
2
2
a −u
2a
u−a
Z
du
1
u−a
20.
=
+C
ln
u2 − a2
2a
u+a
16.
√
Z p
p
up 2
a2
a2 + u2 du =
a + u2 +
ln u + a2 + u2 + C
2
2
Z
p
p
u 2
2
2
2
u a + u du
(a + 2u2 ) a2 + u2 −
=
8
p
a4
ln u + a2 + u2 + C
8
√
√
Z
p
a + a2 + u2
a 2 + u2
2
2
du = a + u − a ln
+C
u
u
√
Z √ 2
p
a + u2
a 2 + u2
du = −
+ ln u + a2 + u2 + C
2
u
u
Z
p
du
√
= ln u + a2 + u2 + C
a 2 + u2
Z
p
up 2
a2
u2 du
√
=
a + u2 −
ln u + a2 + u2 + C
2
2
a 2 + u2
√
1
du
a2 + u2 + a
√
+C
27.
= − ln
a
u
u a2 + u2
√
Z
du
a 2 + u2
√
28.
=−
+C
2
2
2
a2 u
Z u a +u
u
du
+C
29.
= √
2
2
3/2
2
(a + u )
a a2 + u2
Derivatives of y = xm :
Logarithm & Exponential Properties:
ln(a
a·b) = ln(a) + ln(b)
ln
= ln(a) − ln(b)
b
ln(ak ) = k · ln(a)
eln(x) = x
ln(ex ) = x
ea
ea · eb = ea+b
= ea−b
eb
eC+x = eC · ex = C1 ex
y ′ = mxm−1
y ′′ = (m2 − m)xm−2
y ′′′ = (m3 − 3m2 + 2m)xm−3
y (4) = (m4 − 6m3 + 11m2 − 6m)xm−4
Z
Laplace Transform Properties:
′
L{y } = sY (s) − y(0)
L{y ′′ } = s2 Y (s) − sy(0) − y ′ (0)
L{y ′′′ } = s3 Y (s) − s2 y(0) − sy ′ (0) − y ′′ (0)
Laplace Transform Table:
f (t)
F (s)
1
1
s
n!
tn
e
at
sin(kt)
cos(kt)
sinh(kt)
cosh(kt)
sn+1
1
s−a
k
2
s + k2
s
2
s + k2
k
2
s − k2
s
2
s − k2
Derivative Rules:
f
f′
xn
nxn−1
un
nun−1 · u′
ex
ex
eu
eu · u′
au
au ln(a) · u′
1
x
u′
u
ln(x)
ln(u)
Inverse Laplace Table:
F (s)
1
s
1
sn
1
s−a
1
s2 + k 2
s
s2 + k 2
1
2
s − k2
s
s2 − k 2
f (t)
1
tn−1
(n − 1)!
eat
1
sin(kt)
k
cos(kt)
1
sinh(kt)
k
cosh(kt)