Table of some basic integrals
n
 x dx 
x
1 n 1
x  C ; n  1
n 1
x
 a dx 
x
x
 e dx e  C
1
dx   dx  ln | x | C
x
ax
 C ; a  0, a  1
ln a
 sin xdx   cos x  C
1
 tan xdx   ln | cos( x) | C
a
2
1
1
 x
dx  arc tan    C
2
x
a
a
1
1 x
1
 1 x
2
2
x
dx  tan x  C ; |x|<1
1
 cosh
 sinh xdx  cosh x  C

1
 cos
2
x

dx  ar sinh x  C  ln( x  1  x 2 )  C
2
1
 1 x
dx  ar tanh x  C ; |x|<1
1
 ( x  a)
dx  tanh x  C
n
dx 
1
 C ;(n  2)
(1  n)( x  a) n 1
x
2
1
x2 1
dx  ar cosh x  C ; x>1
dx  ar coth x  C ; |x|>1
ax
a
dx  ln | x 2  b | C
b
2
2
Some integration rules:
g 1 ( b )
b
Substitution:
 f ( x)dx  
a

f (ax  b)dx 
f ( g (t )) g (t )dt with x  g (t ); t  g 1 ( x) (inverse function)
1
g (a)
1
F (ax  b)  C mit F ( x)  f ( x) (linear substitution)
a
Integration by parts:
 f ( x) g ( x)dx  f ( x)  g ( x)   f ( x) g ( x)dx
Complex numbers: z  a  bi  x  iy  r (cos   i sin  )  r  e
i
i
Be w  r  e : equation z  w has n solutions: zk 
n
n
r e
  2k 
i 

n n 
; k  0,1, 2,..., n  1
 f  gf   fg  d
f ( x)
;
ln[ f ( x)] 
 
2
dx
f ( x)
g
g
Some differentiation rules: 
Function y = f(x) be invertible: x = g(y)=f-1(y) is the inverse function: then:
g ( y ) 
1
1
; or ( f 1 )( y ) 
f ( x)
f ( f 1 ( y ))

Hyperbolic functions: sinh( x)  0,5  e  e
x
x

; cosh( x)  0,5   e x  e x 
Taylor series expansion: f ( x) 


n0
Some power series: e x 
f ( n ) ( x0 )
n
 x  x0  ;
n!

xn
(1) n 1 n
;
ln(1

x
)

x for |x|<1 ;


n
n 0 n !
n 1

Convergence of series:
an1
| 1 , then the series converges absolutely
an
Ratio criteria: If
lim |
Root criteria: If
lim n | an |  1 , then the series converges absolutely
n 
n 
Ordinary Differential Equations:
Separable ODE:
dy
1
 y  f ( x)  g ( y ) 
dy   f ( x)  dx

dx
g ( y)
Linear ODE of 1st order: y  f ( x)  y  g ( x)

solution of the homogeneous DE: yh ( x)  C  e
f ( x ) dx
 C  h( x )
Variation of constant: y ( x)  C ( x)  h( x) with C ( x)  
g ( x)
dx  K
h( x )
y  ay  by  g ( x)
Linear ODE of 2nd order and constant coefficients:
Characteristic equation:  2  a  b  0
Solution yh of the homogeneous DE: depends on roots:
Two real roots 1 , 2 : yh  C1  e
1 x
 C2  e2 x
Double real root   1  2 : yh   C1  C2 x   e
x
x
conjugated complex roots: 1,2    i : yh  e
 A sin( x)  B cos( x) 
Particular solution y p : e.g. approach with type of right hand side
e.g.: g ( x)  Pn ( x)  ecx sin(ax) and c+ia is no solution of the characteristic equation
cx
 solution approach: y p ( x)  e  (Qn ( x)sin(ax)  Rn ( x) cos(ax))
with Pn, Qn, Rn : polynomial of grade n
General solution of ODE: y  yh  y p
Systems of ODE: two linear ODE of 1st order:
Characteristic equation:
y1  a11 y1  a12 y2  g1 ( x)
y2  a21 y1  a22 y2  g 2 ( x)
 2  (a11  a22 )  (a11a22  a12 a21 )  0