Lösungen / Solutions

1

Remove["Global`*"]

a

Integrate[(t - 1) x^(t - 2) + 5  x^5 - 4  x^4 + t x^3 - x^2, 
      x]

b

Integrate[(t - 1) x^(t - 2) + 5  x^5 - 4  x^4 + t x^3 - 
        x^2, {x, -1, 6  t}]

c

Integrate[-Sin[-t + 2]  + Sin[t + 2], {t, -1, 
        6  x}]

Integrate[Sin[t - 2]  + Sin[t + 2], {t, -1, 6  x}]

Integrate[Sin[t - 2]  + Sin[t + 2], {t, -1, 6  x}] // 
      TrigReduce

Hinweis: Log[x]^2=1 *  Log[x]^2, partiell

d

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

e

Integrate[D[Log[x]^2, x] - 1/x, {x, 1, 2}]

f

Integrate[2 (E^((x^2))) x - Sqrt[x], {x, t, t^2}] // 
      Simplify

g

Integrate[2/(y E^y)  -  Cos[1 - Pi y], 
      y]

Integrate[2/(y E^y)  -  Cos[1 - Pi y], {y, t, t/2}]  // 
      Expand

Integrate[2/(y E^y)  -  Cos[1 - Pi y], {y, t, t/2}]  // 
      N

2/(y E^y) hat keine elementare Stammfunktion

2

2 einfache Modelle für Tunell. Welches besser? 
a x^4+b x^2+c,   d Cos[(π/24) x]+f Sin[(π/24) x]^2
Plot, Integral, Min, Max.,  was passiert bei diesen Modellen unangenehmes?

p1[x_] := a x^4 + b x^2 + c

solv1 = 
      Solve[{p1[12] == 0, p1[10] == 5, p1[0] == 6}, {a, b, c}] // 
        Flatten

N[%]

p1[x_] := a x^4 + b x^2 + c /. solv1;

p1[x]

Plot[p1[x], {x, -12, 12}];

Integrate[p1[x], {x, -12, 12}]

Integrate[p1[x], {x, -12, 12}] // N

Solve[D[p1[x], x] == 0, {x}]

Solve[D[p1[x], x] == 0, {x}] // N

p2[x_] := d Cos[e x] + h Sin[e x]^2

gl = {p2[12] == 0, p2[10] == 5, p2[0] == 6}

solv2 = Solve[gl, {d, e, h}]

System überfordert. Probiere einfacheres Modell, dafür aber zuviele Bedingungen:

p3[x_] := d Cos[Pi/24 x] + h Sin[Pi/24  x]^2

p3[0]

p4[x_] := p3[x] /. d -> 6;  p4[x]

solv3 = Solve[p4[12] == 0, {h}] // Simplify // 
        Flatten

p5[x_] := 
      6 Cos[((π x)/24)] + h Sin[((π x)/24)]^2 /. solv3; 
    p5[x]

p5[10] // N

Plot[{p1[x], p5[x]}, {x, -12, 12}, 
        PlotRange -> {-1, 8}];

Modell sehr schlecht wegen Randverhalten! Probiere anderes Modell:

solv3 = Solve[p4[10] == 5, {h}] // Simplify // 
        Flatten

p5[x_] := 
      6 Cos[((π x)/24)] + h Sin[((π x)/24)]^2 /. solv3; 
    p5[x]

N[%]

Plot[p5[x], {x, -16, 16}, 
        PlotRange -> {-1, 7}];

Integrate[p1[x], {x, -12, 12}]

N[%]

Integrate[p5[x], {x, -12, 12}] // N

Integrate[p5[x], {x, -15, 15}] // N

D[p1[x], x]

Solve[D[p1[x], x] == 0, {x}]

Solve[D[p1[x], x] == 0, {x}] // N

Solve[D[p5[x], x] == 0, {x}]

Solve[D[p5[x], x] == 0, {x}] // N

Beide Modelle mit Sinus und Cosinus problematisch! Nächster Schritt: Faktor k einbeziehen!

3

Zylinder mit selbem Radius?

Remove[f, t, r, f1]

f[x_] := x^3 ;  f1[y_] := y^(1/3)

res  =  Pi Integrate[f1[y]^2, {y, 0, t}]

Pi Integrate[f1[y]^2, {y, 0, t}] // N

Solve[res == r^2 Pi t, {r}]

Zylinder mit selbem Mantel?

2  Pi NIntegrate[
          Evaluate[f1[y] Sqrt[1 + (D[f1[y], y])^2]], {y, 0, 2}] // 
      N

NSolve[13.319247343855286 == 2 r Pi (2 - 0), {r}]

NSolve[
      NIntegrate[Evaluate[f1[y] Sqrt[1 + (D[f1[y], y])^2]], {y, 0, 2}] == 
        r (2 - 0), {r}]

4

Min, Max, WP

f[x_] := x^3 + 3  x^2 + 1  x + 1

Integrate[f[t], {t, 0, x}]

D[f[x], x]

Solve[
      Evaluate[Integrate[f[t], {t, 0, x}] == D[f[x], x]], {x}]

Solve[Evaluate[Integrate[f[t], {t, 0, x}] == D[f[x], x]], {x}] // 
        N // Chop

Plot[f[x], {x, -5, 5}];

fq[x_, q_] := x^3 + q x^2 + 1  x + 1

D[fq[x, q], {x, 1}]

Solve[1 + 2 q x + 3 (x^2) == 0, {q}]

Solve[((-1 - 3 (x^2))/(2 x)) == 0, {x}]

Plot[((-1 - 3 (x^2))/(2 x)), {x, -4, 5}];

D[((-1 - 3 (x^2))/(2 x)), x]

Solve[D[((-1 - 3 (x^2))/(2 x)), x] == 0, {x}]

5

Remove[a, f, f1, f2, f3]

f1[x_] := x^3; f2[x_] := -a (x - 10)^2 + 10;

f3[x_] := f2[x] /. a -> 0.05; {f1[x], f3[x]}

Plot[{f1[x], f3[x]}, {x, 0, 10}];

{f1[t] == f2[t], 
      Integrate[f1[x], {x, 0, t}] == Integrate[f2[x], {x, t, 10}]}

Solve[
      Evaluate[{f1[t] == f2[t], 
          Integrate[f1[x], {x, 0, t}] == Integrate[f2[x], {x, t, 10}]}], {a, 
        t}]

Evaluate[{f1[t] == f2[t], 
        Integrate[f1[x], {x, 0, t}] == 
          Integrate[f2[x], {x, t, 10}]}]

solv6 = 
      Solve[Evaluate[{f1[t] == f2[t], 
              Integrate[f1[x], {x, 0, t}] == 
                Integrate[f2[x], {x, t, 10}]}], {a, t}] // N

Plot[{f1[x], f2[x] /. solv6[[3]]}, {x, 0, 10}];

Plot[{f1[x], f2[x] /. solv6[[4]]}, {x, 0, 10}];