Differential Equations with Boundary-Value Problems,7th Edition.pdf

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
d
7. Power: xn = nxn 1 8. Power: [ g(x)]n = n[ g(x)]n 1 g
(x)
dx dx
Functions
Trigonometric:
d d d
9. sin x = cos x 10. cos x =
sin x 11. tan x = sec2 x
dx dx dx
d d d
12. cot x =
csc2 x 13. sec x = sec x tan x 14. csc x =
csc x cot x
dx dx dx
Inverse trigonometric:
d
1 d
1 d
1
15. sin 1 x = 16. cos 1 x =
17. tan 1 x =
2
dx 1
x2 dx 1
x2 dx 1 + x
d
1 d
1 d
1
18. cot 1 x =
19. sec 1 x = 20. csc 1 x =
2
dx 1 + x dx x x2
1 dx x x2
1
Hyperbolic:
d d d
21. sinh x = cosh x 22. cosh x = sinh x 23. tanh x = sech2 x
dx dx dx
d d d
24. coth x =
csch2 x 25. sech x =
sech x tanh x 26. csch x =
csch x coth x
dx dx dx
Inverse hyperbolic:
d
1 d
1 d
1
27. sinh 1 x = 28. cosh 1 x = 29. tanh 1 x =
2
dx x2 + 1 dx x2
1 dx 1
x
d
1 d
1 d
1
30. coth 1 x = 31. sech 1 x =
32. csch 1 x =
2
dx 1
x dx x 1
x2 dx x x2 + 1
Exponential:
d d
33. ex = ex 34. bx = bx (ln b)
dx dx
Logarithmic:
d 1 d 1
35. ln x = 36. log x =
dx x dx b x(ln b)
BRIEF TABLE OF INTEGRALS
n1
n u 1
1. udu z Cn,1 2. du ln u C
³ n 1 ³ u
3. edueuu  C 4. aduuu 1 a C
³ ³ ln a
5. ³sinudu  cos u  C 6. ³ cosudu sin u C
7. ³sec2 udu tan u C 8. ³ csc2 udu  cot u  C
9. ³secuuduuC tan sec  10. ³ cscuuduuC cot  csc 
11. ³ tanudu  ln cos u  C 12. ³ cotudu ln sin u C
13. ³secudu ln sec u tan u C 14. ³ cscudu ln csc u cot u C
15. ³uuduuuuCsin sin cos  16. ³os cos sin 
17. 2 udu 11 u u  C 18. 2 udu 11 u u  C
³sin24 sin 2 ³ cos24 sin 2
19. ³ tan2 udu tan u u C 20. ³ cot2 udu  cot u  u  C
21. 32udu 1   u u