Lösungen / Solutions

1

a

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

b

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

c

Integrate[1/((x - 5) (x - 3) (x + 6)), x]

d

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

2

a

Integrate[-(1/((Sqrt[1 - (x^2)]) ArcSin[x]^2)), x]

D[1/(ArcSin[x]), x]

b

Integrate[Log[x^2 + 1] - (ArcTan[x]), x] // Simplify

c

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

D[Log[x^2 + 1] - (ArcTan[x]), x] // Together

d

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

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

e

Integrate[-2 a x Sin[b + a (x^2)], x] // Simplify

D[Cos[a x^2  +  b], x]

f

Integrate[a 1/Cos[b + a x] Tan[b + a x], 
        x]//Simplify

D[1/Cos[a x  +  b], x] /.  {Sec -> 1/Cos}

g

Integrate[-a  Sin[b + a x], 
        x]//Simplify

D[Cos[a x  +  b], x]

h

Integrate[-2 a Cos[b + a x] Sin[b + a x], 
        x]//Simplify

D[Cos[a x  +  b]^2, x]

i

Integrate[4 a Cos[b + a x]^3 Sin[b + a x], 
        x]//Simplify

D[-Cos[a x  +  b]^4, x]

j

Integrate[
        2 a Cos[x] Cos[b + a x] Sin[b + a x] - 
          Sin[x] Sin[b + a x]^2, x]//Simplify

D[Sin[a x  +  b]^2 Cos[x], x]

k

Integrate[((2 a Cos[x] Log[b + a x])/(b + a x)) - 
          Log[b + a x]^2 Sin[x], x]//Simplify

D[Log[a x  +  b]^2 Cos[x], x]

l

Integrate[
        a + a Log[b + a x] + ((2 a Log[b + a x])/(b + a x)), 
        x]//Simplify

D[Log[a x  +  b]^2  + (a x  +  b) Log[a x  +  b], 
      x]

3

a

Integrate[E^x x, x]

b

Integrate[E^x x^2, x]

c

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

d

Integrate[(a x +  b) Sin[a x + b], x]

e

Integrate[(a x +  b) Log[a x + b], x]

f

Integrate[(a x +  b)^2  Cosh[a x + b], x]