Aufgabe 1

Integrate[x*Sin[x],x]

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

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

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

Integrate[s*E^s^2,s]

Integrate[t^2*E^t,t]

Aufgabe 3

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

Ableitungen

f'[x]
f''[x]
f'''[x]
f''''[x]

Nullstellen

Solve[f[x]==0,x]

Nullstellen bei x1=-2, x2 = 0, x3=1

Extremwerte

Solve[f'[x]==0,x]

N[%]

f''[(1/3) (-1-(Sqrt[7]))]

%<0

FullSimplify[f[(1/3) (-1-(Sqrt[7]))]]

N[%]

Maximum bei ((1/3) (-1-(Sqrt[7])),(2/27) (10+7 (Sqrt[7])))

f''[(1/3) (-1+(Sqrt[7]))]

% >0

FullSimplify[f[(1/3) (-1+(Sqrt[7]))]]

N[%]

Minimum bei (((1/3) (-1-(Sqrt[7])),(2/27) (10+7 (Sqrt[7])))

Wendepunkte

Solve[f''[x]==0,x]

f'''[-1/3]

f[-1/3]

Wendepunkt bei (-1/3, 20/27)

f'[-1/3]

FullSimplify[Solve[-7/3*(-1/3)+d==20/27,d]]

Wendetangente y = -7/3 x -1/27

Graph

Plot[{f[x],-7/3 x-1/27},{x,-3,3}]