,REVIEW OF DIFFERENTIATION
Rules
d d
1. Constant: c=0 2. Constant Multiple: cf (x) = c f (x)
dx dx
d d
3 . Sum: [ f (x) ± g(x)] = f (x) ± g (x) 4. Product: f (x) g(x) = f (x) g (x) + g(x) f (x)
dx dx
d f (x) g(x)f (x) f (x) g (x) d
5. Quotient: = 6. Chain: f ( g(x)) = f ( g(x)) g (x)
dx g(x) [ g(x)]2 dx
d n 1 d 1
7. Power: x = nx n 8. Power: [ g(x)]n = n[ g(x)]n g (x)
dx dx
Functions
Trigonometric:
d d d
9. sin x = cos x 10. cos x = sin x 11. tan x = sec 2 x
dx dx dx
d d d
12. cot x = csc 2 x 13. sec x = sec x tan x 14. csc x = csc x cot x
dx dx dx
Inverse trigonometric:
d 1 1 d 1 1 d 1 1
15. sin x= 16. cos x= 17. tan x=
dx 1 x 2 dx 1 x 2 dx 1 + x2
d 1 1 d 1 1 d 1 1
18. cot x= 2
19. sec x= 20. csc x=
dx 1+ x dx x x 2
1 dx x x2 1
Hyperbolic:
d d d
21. sinh x = cosh x 22. cosh x = sinh x 23. tanh x = sech 2 x
dx dx dx
d d d
24. coth x = csch 2 x 25. sech x = sech x tanh x 26. csch x = csch x coth x
dx dx dx
Inverse hyperbolic:
d 1 1 d 1 1 d 1 1
27. sinh x= 28. cosh x= 29. tanh x=
dx x +12 dx x 2
1 dx 1 x2
d 1 1 d 1 1 d 1 1
30. coth x= 2
31. sech x= 32. csch x=
dx 1 x dx x 1 x2 dx x x2 + 1
Exponential:
d x d x
33. e = ex 34. b = bx (ln b)
dx dx
Logarithmic:
d 1 d 1
35. ln x = 36. log b x =
dx x dx x(ln b)
,BRIEF TABLE OF INTEGRALS
u n 1 1
1.
³u n du
n 1
C , n z 1 2.
³ u du ln u C
1
³ e du e C ³ a du ln a a C
u u u u
3. 4.
5.
³ sin u du cos u C 6.
³ cos u du sin u C
³ sec u du tan u C ³ csc u du cot u C
2 2
7. 8.
9.
³ sec u tan u du sec u C 10.
³ csc u cot u du csc u C
11.
³ tan u du ln cos u C 12.
³ cot u du ln sin u C
13.
³ sec u du ln sec u tan u C 14.
³ csc u du ln csc u cot u C
15.
³ u sin u du sin u u cos u C 16.
³ u cos u du cos u u sin u C
³ sin u du u sin 2u C ³ cos u du u sin 2u C
2 1 1 2 1 1
17. 2 4
18. 2 4
³ tan u du tan u u C ³ cot u du cot u u C
2 2
19. 20.
³ sin u du 2 sin u cos u C ³ cos u du 2 cos u sin u C
3 1 2 3 1 2
21. 3
22. 3
³ tan u du tan u ln cos u C ³ cot u du cot u ln sin u C
3 1 2 3 1 2
23. 2
24. 2
³ sec u du sec u tan u ln sec u tan u C ³ csc u du csc u cot u ln csc u cot u C
3 1 1 3 1 1
25. 2 2
26. 2 2
sin( a b)u sin( a b)u sin(a b)u sin(a b)u
27.
³ sin au cos bu du 2(a b) 2(a b) C 28.
³ cos au cos bu du 2(a b) 2(a b) C
e au eau
29.
³eau sin bu du
a 2 b2
a sin bu b cos bu C 30.
³eau cos bu du
a2 b2
a cos bu b sin bu C
31.
³ sinh u du cosh u C 32.
³ cosh u du sinh u C
³ sech u du tanh u C ³ csch u du coth u C
2 2
33. 34.
35.
³ tanh u du ln(cosh u) C 36.
³ coth u du ln sinh u C
³ ln u du u ln u u C ³ u ln u du u ln u u C
1 2 1 2
37. 38. 2 4
1 u 1
³ a u du sin a C ³ a u du ln u a u
1
39. 40. 2 2
C
2 2 2 2
u 2 a2 u u 2 a2
41.
³ a 2 u 2 du
2
a u2
2
sin 1 C
a
42.
³ a 2 u 2 du
2
a u 2 ln u a 2 u 2 C
2
1 1 u 1 1 au
43.
³a 2
u2
du
a
tan 1 C
a
44.
³a 2
u2
du ln
2a a u
C
Note: Some techniques of integration, such as integration by parts and partial fractions, are
reviewed in the Student Resource and Solutions Manual that accompanies this text.
, TABLE OF LAPLACE TRANSFORMS
f (t) ᏸ{ f (t)} F(s) f (t) ᏸ{ f (t)} F(s)
1 k
1. 1 20. e at sinh kt
s (s a)2 k2
1 sa
2. t 21. e at cosh kt
s2 (s a)2 k2
n! 2ks
3. t n , n a positive integer 22. t sin kt
sn1 (s2 k2)2
Bs
s2 k2
4. t 1/2 23. t cos kt
(s2 k2)2
1 2 ks2
5. t 1/2 24. sin kt kt cos kt
2s3/2 (s k2)2
2
( 1) 2 k3
6. t a , a 1 25. sin kt kt cos kt
s1 (s k2)2
2
k 2 ks
7. sin kt 26. t sinh kt
s2 k2 (s2 k2)2
s s2 k2
8. cos kt 27. t cosh kt
s2 k2 (s2 k2)2
2k 2 eat ebt 1
9. sin2 kt 28.
s(s2 4k2) ab (s a)(s b)
s2 2k2 aeat bebt s
10. cos2 kt 29.
s(s2 4 k2) ab (s a)(s b)
1 k2
11. e at 30. 1 cos kt
sa s(s k2)2
k k3
12. sinh kt 31. kt sin kt
s k2
2
s (s k2)
2 2
s a sin bt b sin at 1
13. cosh kt 32.
s2 k2 ab (a2 b2) (s2 a2)(s2 b2)
2k2 cos bt cos at s
14. sinh2 kt 33.
s(s2 4k2) a2 b2 (s2 a2)(s2 b2)
s2 2k2 2 k2s
15. cosh2 kt 34. sin kt sinh kt
s(s2 4k2) s4 4k4
1 k(s2 2 k2 )
16. te at 35. sin kt cosh kt
(s a)2 s4 4k4
n! k(s2 2k2 )
17. t n e at , n a positive integer 36. cos kt sinh kt
(s a)n1 s4 4k4
k s3
18. e at sin kt 37. cos kt cosh kt
(s a)2 k2 s 4k4
4
1s2 k2
sa 1
19. e at cos kt 38. J 0 (kt)
(s a)2 k2
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