Lösungen / Solutions /I1S2Ue/Ex10

L = Lektion/ Leçon   -- Sprechstunde / heures de consultation
W = Work / Arbeit / Travail
A  =Anhang/  Annexe / Appendice

Ueb/Ex 1      W

a

Integrate[5  x^5 - 4  x^α + 1/2  x^2 - 4  x + 9, x]

b

Integrate[x^2, {x, 0, t^2}]

c

Integrate[
      1/ω Cos[ω t + β], {t, 0, Pi}]

Integrate[1/ω Cos[ω t + β], {t, 0, Pi}] // 
      TrigReduce

Integrate[1/ω Cos[ω t + β], {t, 0, Pi}] // 
      Simplify

d

Integrate[y E^y, {y, -1, 1}]

e

Integrate[x E^(x^2), {x, 0, 1}]

f

Integrate[
      Evaluate[D[Log[e^(x^2) + 2  x Cos[3  x - 2]^2], x]], {x, a , 
        t}]

g

Integrate[((2 (E^((Sqrt[1 - 2 (x^2)]))) x)/(Sqrt[1 - 2 (x^2)])), {x, 0, 1/2}]

h

Integrate[1/(E^x* x^2) - Log[1/x]/E^x, x]

Integrate[E^-x* x^-2 - Log[1/x]*E^-x, x]

Ueb/Ex 2      W

a

Remove["Global`*"]

f5[x_] := -a x^2  +  b;
Solve[f5[x] == 0, {x}] // Flatten

tr[x1_] := (2 ((Sqrt[b])/(Sqrt[a])) +  2  x1)/2 f5[x1];  
    tr[x1] // Expand

Solve[tr[x1] == 0, {x1}]

b

solv2b = Solve[f5[x] == 0, {x}] // Flatten

x1 = x /. solv2b[[1]]; 
    x2 = x /. solv2b[[2]]; {x1, x2}

int = 1/4  Integrate[f5[x]^2 Pi, {x, x1, x2}]

b1

{f5[0], f5[((Sqrt[b])/(Sqrt[a]))]}

Solve[f5[x] == y, {x}]

int1 = 
      Integrate[((( (Sqrt[b - y]))/(Sqrt[a])))^2 Pi, {y, 0, 
          b}]

c

int /. {a -> 1/4, b -> 36}

N[%]

Ueb/Ex 3      W

a

Remove["Global`*"]

f[x_] := a x^2  +  b; f[x]

f[0]

{f[4] == t, f[0] == 0}

{f[4] == t, b == 0}

solv1 = 
      Solve[{f[4] == t, b == 0}, {a, b}] // Flatten

f1[x_, t_] := a x^2  +  b /. solv1; f1[x, t]

{f[4] == t, f[0] == 4}

{f[4] == t, b == 4}

solv2 = 
      Solve[{f[4] == t, b == 4}, {a, b}] // Flatten

f2[x_, t_] := a x^2  +  b /. solv2; f2[x, t]

b

Integrate[f2[x, t] - f1[x, t], {x, -4, 4}]

c

Inhalt unabhängig von t / Contenu indépendent de t

Ueb/Ex 4      W

a

Remove["Global`*"]

p[x_] := x (x - Pi)/(Pi/2 (Pi/2 - Pi))

Plot[{Sin [x], p[x]}, {x, 0, Pi}];

Integrate[Evaluate[Sqrt[1 + D[p[x], x]^2]], {x, 1, Pi}]

l1 = 
      NIntegrate[Evaluate[Sqrt[1 + D[p[x], x]^2]], {x, 1, Pi}]

b

Integrate[Evaluate[Sqrt[1 + D[Sin[x], x]^2]], {x, 1, Pi}]

l2 = 
      NIntegrate[Evaluate[Sqrt[1 + D[Sin[x], x]^2]], {x, 1, Pi}]

l1/l2

Ueb/Ex 5      W

a

Plot[E^(-x^2), {x, -Pi, Pi}];

solv5 = Solve[y == E^(-x^2), {x}] // Flatten

h5 := x /. solv5[[2]];  h5

Integrate[h5^2  Pi, {y, 0, 1}]

Integrate[(I (Sqrt[Log])[y])^2  Pi, {y, 0, 1}]

b

solv5b = 
      FindRoot[Integrate[E^(-x^2 * a), {x, -Pi, Pi}] == 
          1, {a, 1}]

a1 = a /. solv5b[[1]]

Plot[{E^(-x^2), E^(-x^2 a1)}, {x, -Pi, 
          Pi}];