Diverse Integrale mit Lösungen (ohne Integrationskonstanten)

Die Lösungen wurden kontrolliert mit:

    |\^/|     Maple 7 (SUN SPARC SOLARIS)
._|\|   |/|_. Copyright (c) 2001 by Waterloo Maple Inc.
 \  MAPLE  /  All rights reserved. Maple is a registered trademark of
 <____ ____>  Waterloo Maple Inc.
      |       Type ? for help.


56)
> int(1/(3*x^2-2*x+4),x);
                           1/2                         1/2
                    1/11 11    arctan(1/22 (6 x - 2) 11   )

57)
> int((6*x-7)/(3*x^2-7*x+11),x);
                                    2
                              ln(3 x  - 7 x + 11)

58)
> int(1/(2-3*x-4*x^2)^(1/2),x);
                                         1/2
                       1/2 arcsin(8/41 41    (x + 3/8))

59)
> int(1/(1+x+x^2)^(1/2),x);
                                       1/2
                          arcsinh(2/3 3    (x + 1/2))

60)
> int((2*a*x + b)/(a*x^2 + b*x + c)^(1/2),x);
                                   2           1/2
                             2 (a x  + b x + c)

61)
> int((x+3)/(4*x^2+4*x+3)^(1/2),x);
                     2           1/2                1/2
             1/4 (4 x  + 4 x + 3)    + 5/4 arcsinh(2    (x + 1/2))

62)
> int(x*exp(x),x);
                               x exp(x) - exp(x)

63)
> int(x*ln(x),x);
                                  2              2
                             1/2 x  ln(x) - 1/4 x

64)
> int(x*sin(x),x);
                               sin(x) - x cos(x)

65)
> int(ln(x),x);
                                  x ln(x) - x

66)
> int(arcsin(x),x);
                                               2 1/2
                           x arcsin(x) + (1 - x )

67)
> int(ln(x-1),x);
                           ln(x - 1) (x - 1) - x + 1

68)
> int(x^n*ln(x),x);
                 x^(n + 1)ln(x)/(n + 1) - x^(n + 1)/(n + 1)^2

69)
> int(x*arctan(x),x); #+1,-1 Trick
                        2
                   1/2 x  arctan(x) - 1/2 x + 1/2 arctan(x)

* 70)
> int(arcsin((x/(x+1))^(1/2)),x);
                                 1/2 /  1  \1/2   1/2           1/2
                                x    |-----|    (x    - arctan(x   ))
                /  x  \1/2           \x + 1/
         arcsin(|-----|   ) x - -------------------------------------
                \x + 1/                      /  x  \1/2
                                             |-----|
                                             \x + 1/
#mein Versuch
diff(arcsin((x/(x+1))^(1/2))*x - 2*(x+1)^(5/2)/5 + 2*(x+1)^(3/2)/3,x);
* 71)
> int(arcsin(x)/x^2,x);
                        arcsin(x)                1
                      - --------- - arctanh(-----------)
                            x                     2 1/2
                                            (1 - x )

82)
> int(x^(1/2)/(x^(3/2)+1),x);
                                       3/2
                               2/3 ln(x    + 1)

83)
> int((x^(3/2)-x^(1/3))/(6*x^(1/4)),x);
                                               13
                                               --
                                   9/4         12
                             2/27 x    - 2/13 x

* 86)
> int(sin(x)^3,x);
                                   2
                       - 1/3 sin(x)  cos(x) - 2/3 cos(x)

 diff(-sin(x)^2*cos(x)-2*cos(x)^3+cos(2*x)/2,x);

89)
> int(cos(x)^2,x);
                                1/2 cos(x) sin(x) + 1/2 x

# cos(x)^2 = ( 1 + cos(2*x) )/2 verwenden
# => 1/4 sin(2*x) + 1/2 x => sin(2*x) = ...

95)
> int(sin(x)/(1+sin(x)),x);
                                2
                          -------------- + 2 arctan(tan(1/2 x))
                          tan(1/2 x) + 1

#Silver Bullet aka Rationalisierungsformel

96)
> int(cos(x)/(1+cos(x))^2,x);
                                                           3
                            1/2 tan(1/2 x) - 1/6 tan(1/2 x)

97)
> int(1/(1+cos(x))^2,x);
                                          3
                            1/6 tan(1/2 x)  + 1/2 tan(1/2 x)