Lösungen / Solutions

1

g[t_] := E^t;    f[x_] := (g[x^4] - 1)/x;   f[x]

a

?Series

Series[g[t], {t, 0, 10}]

(Series[g[t], {t, 0, 10}] - 1)/x   /. t -> x^4

(Series[g[t], {t, 0, 10}] - 1)/x   /. t -> x^4  //  
      Normal

b

fac[0] = 1;  fac[n_] := fac[n - 1] n;  fac[5]

a[m_] := 1/fac[m];   n[m_] := 4  m - 1;  a[n[5]]

q[m]=a[n[m]] / a[n[m]+1] =( 1/1/fac[n[m]]) / ( 1/1/fac[n[m]+1]) 
= fac[n[m]+1]/ fac[n[m]+1] = n[m]+1

q[m_] := m + 1

Limit[q[n[m]], {m -> Infinity}]

c

Integrate[
      Normal[(Series[g[t], {t, 0, 10}] - 1)/x   /. t -> x^4 ], 
      x]

2

a

f[x_] := 1/(x - 10);  f[x]

-1/10  * 1/(1 - x/10)  //  Simplify

-1/10  * 1/(1 - q)  //  Simplify

-1/10  * 
        1/(1 - q)  /.{q -> x/10}

-1/10  * 1/(1 - q)   /. {q -> x/10}   // 
      Simplify

f1[
          q_] := -1/
            10  *(1 + q + q^2 + q^3 + q^4 + q^5 + 
              q^6) ;
f1[x]

f1[q]  /. {q -> x/10}   //  Expand

b

Series[f[x], {x, 0, 10}]

r = 
      Limit[(1/10^n)/(1/10^(n + 1)), {n -> 
            Infinity}]

3

f[x_] := x^2;  f[1]

g[
        t_] := (f[x0] - f[-1])/(x0 - (-1)) t + b  //  
        Simplify;  g[t]

solv = Solve[g[-1] == 1, {b}] //  Flatten

g[t_, 
        x0_] := (f[x0] - f[-1])/(x0 - (-1)) t + b  /. 
          solv  //  Simplify;  g[t, x0]

A2[x0_] := Integrate[g[t, x0]  - f[t], {t, -1, x0}];  
    A2[x0]

Aunten = Integrate[f[t], {t, -1, 1}]

Aoben = 2*1  -  Aunten

A1[x0_] := Aoben - A2[x0];   A1[x0]

A1[x0] ==  A2[x0]

solv2 = Solve[  A1[x0] ==  A2[x0], {x0}] // Flatten

x0 /. solv2[[1]]

N[%]

4

f[x] := Sqrt[x];   f[x]

V = Integrate[f[x]^2 Pi, {x, 0, 1}]

5

a

Integrate[Log[x], {x, 1, 2}]

N[%]

b

Integrate[x*(Sqrt[x + 1])^3, x]

Factor[%]

c

Integrate[x*Sin[5  x + 7], x]

Integrate[x*Sin[5  x + 7], x] // Simplify

d

Integrate[Tan[x], x]

e

Integrate[4  x/((x + 4) (x + 1)), x]

6

Integrate[Sin[Sin[x]], {x, -1, 1}]