Lösungen E+M1 S2 AnUe 1

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

Ueb/Ex 1      W

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

(* Riemannsche Summe *)
Sum[1/n (k/n)^2,{k,1,n}]

%//Apart

A = Limit[Sum[1/n (k/n)^2,{k,1,n}],n->Infinity]

Ueb/Ex 2      W

a

Remove["Global`*"]

f[x_]:=E^(-x^2)

Integrate[f[x], x]

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

N[%]

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

A = Limit[2 Sum[2/n f[2 k/n],{k,1,n}],n->Infinity]

n = 10^2;
2 Sum[N[2/n f[2 k/n]],{k,1,n}]

n = 10^3;
2 Sum[N[2/n f[2 k/n]],{k,1,n}]

n = 10^4;
2 Sum[N[2/n f[2 k/n]],{k,1,n}]

n = 10^5;
2 Sum[N[2/n f[2 k/n]],{k,1,n}]

n = 10^5;
2 Sum[N[2/n f[2 k/n]],{k,1,n}]

n = 4 10^5;
2 Sum[N[2/n f[2 k/n]],{k,1,n}]

b

Series[f[x],{x,0,100}]

Normal[Series[f[x],{x,0,100}]]

Integrate[Normal[Series[f[x],{x,0,10}]],{x,-2.,2.}]

Integrate[Normal[Series[f[x],{x,0,20}]],{x,-2.,2.}]

Integrate[Normal[Series[f[x],{x,0,33}]],{x,-2.,2.}]

Integrate[Normal[Series[f[x],{x,0,35}]],{x,-2.,2.}]

Ueb/Ex 3      W

a

Integrate[x^100 - x^99, x] + c

b

Sin[x]^2 + Cos[x]^2 // Simplify

Integrate[E^-x  -  Cos[x]  +  1  -  1/x^2 , x] + 
      c

c

Integrate[Cosh[x] - Sinh[x] , x] + c

d

Integrate[x^(1/3) , x] + c

e

Apart[1/(x^2-1)]

Integrate[1/(x^2 - 1) , x] + c

1/2 (Integrate[1/(x-1) ,x] + Integrate[1/(x+1) ,x])+c

Ueb/Ex 4      W

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

N[%]

Ueb/Ex 5     W

a

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

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

N[%]

b

Integrate[Cosh[x],x]+c

Integrate[Cosh[x],{x,-1,1}]

N[%]

c

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

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

N[%]

d

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

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

Integrate[Sin[4x-7]^2+Cos[4x-7]^2,{x,1,2}]

e

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

Integrate[1/Cos[x]^2,{x,0,1}]

N[%]

f

Integrate[ Cos[ω x  +  φ], x]

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

Integrate[ Cos[ω x + φ],{x,1,2}]//TrigReduce

g

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

Integrate[E^(4x-3)-2 Log[4x+3]/(x+3/4),{x,1,4}]

N[%]

h

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

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

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

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

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

Limit[Integrate[1/(2x-3)-10 x^20+x^40,{x,1,3/2-a}]
+Integrate[1/(2x-3)-10 x^20+x^40,{x,3/2+a,2}],a->0]

N[%]

Ueb/Ex 6     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}]

Ueb/Ex 7     W

a

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

Solve[Integrate[x^5, {x, 3, t}] == 10, {t}]

N[%]

b

Integrate[1/(2  x + 1), {x, 4, 6}]

N[%]

c

Integrate[x^2 Sin[x], {x, 0 , Pi}]

N[%]

d

Integrate[1/(4  x^2 - 1), {x, 2, 4}]

Apart[1/(4  x^2 - 1)]

Integrate[Apart[1/(4  x^2 - 1)], {x, 2, 4}]

N[%]

e

Plot[x^5 Cos[x], {x, -4, 4}];

Integrate[x^5 Cos[x], {x, -4, 4}]

f

Integrate[x^2 Cos[4 x^3  + 5], x]

(1/12) Cos[4 (x^3)] Sin[5] + (1/12) Cos[5] Sin[4 (x^3)] // 
      TrigReduce

g

Integrate[Cos[x] E^Sin[x], x]

Integrate[Cos[x] E^Sin[x], {x, 0, Pi}]

Plot[Cos[x] E^Sin[x],{x,0,Pi}];