Lösungen/Solutions/E1S1AnUe/Ex13

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

Ueb/Ex 1      W

Integrate[x^2 - x^3, {x, 0, 1}]

Ueb/Ex 2      W

a

Integrate[x Sin[x^2], x]

b

Integrate[Cosh[x], x]

c

Integrate[-8 x^3  +  4 x^2  - 3 x  +  1  -  2/x, 
      x]

d

Sin[4  x - 7]^2 + Cos[4  x - 7]^2 //  TrigReduce

Integrate[Sin[4  x - 7]^2 + Cos[4  x - 7]^2, x]

e

Integrate[1/Cos[x]^2, x]

f

Integrate[ Cos[ω x  +  φ], x]

Integrate[ Cos[ω x  +  φ], x] // 
      TrigReduce

g

Integrate[E^(4  x - 3) - 2 Log[4  x + 3]/(x + 3/4), 
      x]

h

Integrate[1/(2  x - 3) - 10 x^20 + x^40, x]

Ueb/Ex 3      W

f[x_] := x^2

f'[x0]

t[x0_, x_] := f[x0] + (x - x0) f'[x0];  t[x0, x]

φ[x0_] := ArcTan[(-1)/ f'[x0]];  φ[
      x0]

α[x0_] :=  φ[x0] - 10* 2 Pi /360 ;  α[
      x0]

Remove[nt,  solv,   ntx0]

nt0[x0_, x_] := 
        f[x0] + (x - x0) Tan[φ[x0]];

nt[x0_, x_] := f[x0] + (x - x0) Tan[α[x0]];

nSolv = Solve[nt[x0, 0] == 2, {x0}] // N //  Chop

solv = Solve[nt[x0, 0] == 2, {x0}]

solv = Solve[nt[x0, 0] == 2, {x0}] //  Flatten // N //  
        Chop

solv[[2]]

x1 = x0 /. solv[[2]]

ntx0[x_] := nt[x0, x] /.  solv[[2]] ;   ntx0[x]

p1 = Plot[{f[x], ntx0[x]}, {x, -4, 4}];

p2 = 
        Plot[{f[x], ntx0[x], t[x1, x], nt0[x1, x]}, {x, -4, 
            4}, 
AspectRatio -> 16/8];

solv2 = Solve[{f[x] ==  ntx0[x]}, {x}] // Flatten

x2 = x /. solv2[[2]]

Integrate[ntx0[x] - f[x], {x, x1, x2}]