Derivatives
Integrals
d
[f (u) · g(u)] = [f 0 (u) · g(u) + f (u) · g 0 (u)] u0
dx
#
" 0
d f (u)
f (u) · g(u) − f (u) · g 0 (u) 0
2.
=
u
2
dx g(u)
[g(u)]
1.
Z
1.
Z
2.
Z
d n
3.
[u ] = nun−1 u0
dx
d
1 0
4.
[ln u] =
u
dx
u
3.
Z
4.
d
1
[loga u] =
u0
dx
u ln a
6.
d u
[e ] = [eu ] u0
dx
ua+1
+ C, a 6= −1
a+1
1
du = ln |u| + C
u
eu du = eu + C
au du =
1 u
a +C
ln a
Z
sin u du = − cos u + C
5.
5.
ua du =
Z
cos u du = sin u + C
6.
Z
7.
Z
d u
[a ] = [au ln a] u0
7.
dx
8.
d
[sin u] = [cos u] u0
8.
dx
9.
sec2 u du = tan u + C
csc2 u du = − cot u + C
Z
tan u du = − ln | cos u| + C
Z
9.
10.
d
[cos u] = [− sin u] u0
dx
d
[tan u] = sec2 u u0
dx
cot u du = ln | sin u| + C
10.
Z
sec u du = ln | sec u + tan u| + C
11.
Z
csc u du = − ln | csc u + cot u| + C
12.
d
11.
[cot u] = − csc2 u u0
dx
12.
d
[sec u] = [sec u tan u] u0
dx
d
[csc u] = [− csc u cot u] u0
dx
d
1
√
14.
[arcsin u] =
u0
2
dx
1−u
d
−1
15.
[arccos u] = √
u0
dx
1 − u2
d
1
16.
[arctan u] =
u0
dx
1 + u2
13.
d
−1
17.
[arccot u] =
u0
dx
1 + u2
d
1
√
18.
[arcsec u] =
u0
dx
|u| u2 − 1
19.
d
−1
√
[arccsc u] =
u0
dx
|u| u2 − 1
Z
u
1
du = arcsin + C
2
a
−u
Z
h
i
p
1
√
14.
du = ln u + u2 ± a2 + C
u2 ± a2
Z
1
|u|
1
√
du = arcsec
+C
15.
a
a
u u2 − a2
√
Z
1
−1 a + a2 ± u2
√
ln
+C
16.
du =
a
|u|
u a2 ± u2
Z
1
u
1
17.
du = arctan + C
2
2
a +u
a
a
Z
a + u
1
1
+ C = 1 arctanh u + C
18.
du =
ln
a2 − u2
2a a − u a
a
p
Z p
1
u
2
2
2
2
2
19.
a − u du =
u a − u + a arctan √
+C
2
a2 − u2
Z p
h
i
p
1 p 2
20.
u2 ± a2 du =
u u ± a2 ± a2 ln u + u2 ± a2 + C
2
Z
eau
21.
eau cos bu du = 2
[a cos bu + b sin bu] + C
a + b2
Z
eau
22.
eau sin bu du = 2
[a sin bu − b cos bu] + C
a + b2
13.
√
a2
Laplace Transforms
1. L {f (t)} = F (s)
19. L eat =
1
s−a
2. L {af (t) + bg(t)} = aF (s) + bG(s)
3. L {f 0 (t)} = sF (s) − f (0)
4. L {f 00 (t)} = s2 F (s) − sf (0) − f 0 (0)
n
o
5. L f (n) (t) = sn F (s) − sn−1 f (0) − · · · − f (n−1) (0)
Z
f (τ ) dτ
=
0
n!
(s − a)n+1
21. L {cos kt} =
s
s2 + k 2
k
+ k2
s
23. L {cosh kt} = 2
s − k2
22. L {sin kt} =
t
6. L
20. L tn eat =
F (s)
s
at
7. L e f (t) = F (s − a)
8. L {u(t − a)f (t − a)} = e−as F (s)
Z t
f (τ )g(t − τ ) dτ = F (s)G(s)
9. L
s2
24. L {sinh kt} =
s2
25. L eat cos kt =
0
10. L {tf (t)} = −F 0 (s)
k
− k2
s−a
(s − a)2 + k 2
14. L {1} =
1
s
k
(s − a)2 + k 2
1
1
27. L
(sin
kt
−
kt
cos
kt)
=
2
3
2
2k
(s + k 2 )
t
s
28. L
sin kt =
2
2
2k
(s + k 2 )
1
s2
29. L
(sin kt + kt cos kt) =
2
2k
(s2 + k 2 )
15. L {t} =
1
s2
30. L {u(t − a)} =
26. L eat sin kt =
11. L {tn f (t)} = (−1)n F (n) (s)
Z ∞
f (t)
=
F (σ) dσ
12. L
t
s
1
13. L {f (t), period p} =
1 − e−ps
16. L {tn } =
17. L
1
√
πt
p
e−st f (t) dt
0
n!
e−as
s
31. L {δ(t − a)} = e−as
n
o 1
as 32. L (−1)[|t/a|] (square wave) = tanh
s
2
−as
t
e
33. L (staircase) =
a
s (1 − e−as )
sn+1
18. L {ta } =
Z
1
=√
s
Γ(a + 1)
sa+1
Miscellaneous Functions
1. eix = cos x + i sin x
2. sinh x =
ex − e−x
2
ex + e−x
2
sinh x
4. tanh x =
cosh x
3. cosh x =
Fourier Series

Z

1 L


a
=
f (t) dt
0


L −L



Z
∞ 
a0 X
nπt
nπt
1 L
nπt
+
an cos
+ bn sin
where an =
f (t) =
f (t) cos
dt

2
L
L
L
L

−L
n=1


Z


1 L
nπt


b
=
f (t) sin
dt
 n
L −L
L
5. arctanh u =
1 1 + u ln ,
2
1 − u
−1 < u < 1