Lösungen/ Solutions

1

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

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

<< Graphics`FilledPlot`

p1 = Plot[f[x], {x, -2.3, 2.3}];

p2 = FilledPlot[f[x], {x, -2.0, 2.0}];

Show[p2, p1];

Integrate[f[x], {x, -2, 2}]

2

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

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

p4 = 
        Plot[{Cos[x], -Cos[x] + 1}, {x, -(π/3) - 
              0.3, (π/3) + 0.3}];

Show[p3, p4];

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

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

N[%]

3

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

N[%]

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

N[%]

4

Integrate[2 x^2 Log[x], x]

N[%]

Integrate[2 x^2 Log[x], {x, 3, 5}]

N[%]

5

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

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

N[%]

6

Integrate[2 Sin[3 x  +  4], x]

N[%]

Integrate[2 Sin[3 x  +  4], {x, 0, Pi}]

N[%]

7

Apart[1/((x - 3) (x - 4))]

Integrate[1/((x - 3) (x - 4)), x]

Integrate[1/((x - 3) (x - 4)), {x, 7, 8}]

N[%]