Lösungen/ Solutions

1

f[x_] := a x^2 + b x  +  c;  
    s = Solve[{f[-4] == 0, f[0] == 3/2, f[4] == 0}, {a, b, c} ] // 
        Flatten

N[%]

f[x_] := a x^2 + b x  +  c  /.  s;  f[x]

<< Graphics`FilledPlot`

p1 = Plot[f[x], {x, -4, 4}];

p2 = FilledPlot[f[x], {x, -4, 4}];

Show[p2, p1];

Integrate[f[x], {x, -4, 4}]

2a

Solve[2  Cos[x] == - Cos[x] + 1, {x}]

p3 = 
        FilledPlot[{2  Cos[x], -Cos[x] + 1}, {x, -ArcCos[(1/3)], 
            ArcCos[(1/3)]}];

p4 = 
        Plot[{2  Cos[x], -Cos[x] + 1}, {x, -ArcCos[(1/3)] - 1, 
            ArcCos[(1/3)] + 1}];

Show[p3, p4];

Integrate[2  Cos[x] - (-Cos[x] + 1), x]

Integrate[
      2  Cos[x] - (-Cos[x] + 1), {x, -ArcCos[(1/3)], 
        ArcCos[(1/3)]}]

N[%]

2b

Solve[2  Cos[x] == - Cos[x] + 2, {x}]

N[%]

p3 = 
        FilledPlot[{2  Cos[x], -Cos[x] + 2}, {x, -ArcCos[(1/3)], 
            ArcCos[(1/3)]}];

p4 = 
        Plot[{2  Cos[x], -Cos[x] + 2}, {x, -ArcCos[(1/3)] - 1, 
            ArcCos[(1/3)] + 1}];

Show[p3, p4];

Integrate[2  Cos[x] - (-Cos[x] + 2), x]

Integrate[
      2  Cos[x] - (-Cos[x] + 2), {x, -ArcCos[(2/3)], 
        ArcCos[(2/3)]}]

N[%]

3

Integrate[3 x^5  -  5 x^4  +  6 x^3  -  7 x  +  1, 
      x]

N[%]

Integrate[
      3 x^5  -  5 x^4  +  6 x^3  -  7 x  +  1, {x, 1, 
        2}]

N[%]

4

Integrate[3 x^3 Log[x], x]

N[%]

Integrate[3 x^3 Log[x], {x, 3, 5}]

N[%]

5

Integrate[2  x  x E^(x^2), x]

Integrate[2  x x E^(x^2), {x, 3, 5}]

N[%]

6

Integrate[3 Cos[3 x  +  4], x]

Simplify[%]

N[%]

Integrate[3 Cos[3 x  +  4], {x, 0, Pi}]

N[%]

7

Apart[1/((x - 5) (x - 8))]

Integrate[1/((x - 5) (x - 8)), x]

Integrate[1/((x - 5) (x - 8)), {x, 7, 8}]

N[%]