Integralrechnung

Beispielpool
Integral
001
002
003
004
005
006
007
008
009
010
Error!
Error!
Error!
Error!
Error!
Error!
Error!
Error!
Error!
Error!
5x + 3 ln x – Error! + C
Error! + C
6x + 14 ln(x – 2) + C
Error! ln (3x2 + 4) + C
Error! + Error! + C
Error! + C
Error! + Error! + C
Error! + 6x – 2 ln(x) + 14 ln(x – 2) + C
C – (5x2 + 10x + 13) e–x
C – Error!
011
Error! =
5x – 3 ln(x) – Error! + C
012
Error! =
Error! + C
013
014
015
016
017
018
019
020
021
022
023
024
Error! =
Error! =
Error!
Error!
Error!
Error!
Error!
Error!
Error!
Error!dx
Error!
Error!
C – Error!
C – Error!
Error! + C
C – Error!
5x + 10 ln(x – 2) + C
e2x (1,5x + 1,75) + C
2
(x –12x + 16) ex + C
Error! + C
Error! + C
2 sin(4x – 4) + C
ln(sin(x)) + C
3 ln(x –4) – ln(x + 2) + C
025
Error!
3 ln(2x2 + 4) + C
026
Error!
22 ln(x – 5) – 2 lnx + C
027
Error!
Error! + C
028
Error!
5 e0,2x (2x2 – 20x + 105) + C
029
Error!
4 Error! + C
030
Error!
Error! + C
031
Error! dx =
032
Error!dx =
Error! + c
033
Error!dx =
Error! + c
034
Error!dx =
3 sin(x2) + c
035
Error!dx =
Error! + c
2x2 + 3x + 5 ln(x) + c
036
Error!dx =
Error! + c
037
Error! =
–2x cos(x) + 2 sin(x) + c
038
Error! =
ex (7 – 2x) + c
039
Error! =
c – e–x (3x3 + 14x2 + 28x + 36)
040
Error!
c – Error!
041
Error! =
c – 2 ln(cos(x))
042
Error! =
7x (ln x – 1)
=
043 Error! dx =
044
045
046
047
048
049
050
051
052
Error!dx =
Error!dx =
Error!dx =
Error! dx =
Error!dx =
Error! =
Error! =
Error! =
Error! =
+C
3x4 + 4x + 8 ln(x) + Error!+ c
Error! + C
Error! + C
– 3 cos(4x3) + C
2 ln2(x) + C
Error! + C
C – 5 cos(x) + 5 sin(x)
(25x2 – 265x + 1.365) e0,2x + C
1,5 x2 – 6x + 16 ln(x + 2) + C
3 ln(x + 4) – 3 ln(x) – Error!
053 Error! = 5x5 + 2 cos(2x) + 4 ln(x) + C
054 Error! =
Error!+ C = Error! + C
055 Error! = (3x + 2) 10 e 0,1x – Error! =
(30x + 20) e 0,1x – 300 e 0,1x + C = e 0,1x (30x – 280) + C
056 Error! = 3x2 ( Error!) – Error!dx = Error! + 6x Error! – Error! = Error! + 6x
Error! + Error! = Error!+ C
057 Error! = Error! = Error! – Error!
Error! = Error!  5x + 3 = A(x – 7) + Bx(x – 7) + Cx2  38 = 49C  C =
Error! 3 = –7 A  A = – Error! 43 = A + 8B + 64C  B = Error!
058 Error! = Error! = 2x3 – 12x2 + 96x – 384 ln(x + 4) + C
059
Error! =
060
Error! =
061 Berechnen Sie Error! und vereinfachen Sie das Ergebnis so weit wie möglich.
Error! = 3x2 cos(2 – x) – Error! = 3x2 cos(2 – x) – ( 6x (–sin(2 – x) – Error! = 3x2
cos(2 – x) + 6x sin(2 – x) – 6 cos(2 – x)
062
a)
Berechnen Sie die folgenden Integrale und vereinfachen Sie die Ergebnisse so weit wie möglich:
Error! = 6 sin(3x + 4) + C
b)
c)
063
a)
b)
c)
Error! = Error!
Error! = Error!
=
Error! = 4 ln(3x2 + 4) + C
Berechnen Sie die folgenden Integrale und vereinfachen Sie die Ergebnisse so weit wie möglich:
Error! = 4x cos (3 – x) – Error! = 4x cos(3 – x) + 4 sin(3 – x) + C
Error! = Error! = Error!+ C
Error! = Error! = 3x2 + 30x + 150 ln(x – 5) + C
Error! = Error!
064
Error! = – Error!
065
066 Error! = Error!
067 Error! = Error!
= Error! = Error! = 2 Error!+ C
068 Error! = (2x – 3) Error! – Error!dx = (2x – 3) Error! – Error! + C =
= Error! (–10x +15–2) = Error!
069 Error! = Error! – Error! = Error! – Error! =
= Error!+ C
070 Error! = Error! = Error!+ C
Error! = Error!
A = 2 C = – 0,5 B = 0,5
071 Error! = Error! = Error!
072 Error! = 4x2 – 6x + 3 ln(x) + C
073 Error! = 4 cos(3 – x) + 4e3x + C
074 Error! = Error! = Error! = 4 Error!+ C
075 Error!
= Error!= Error!= 2 ln(5x2 + 3) + C
076 Error!
= Error! – Error! = Error! + Error! + C = Error! + C
077 Error!
= (x2 + 3x) ex – Error! = (x2 + 3x) ex – [ (2x + 3) ex – Error!] = ex (x2 + 3x – 2x – 3 +
2) + C = ex (x2 + x – 1) + C
078 Error!
Error! = Error!  x – 22 = A(x – 4) + B(x + 2) 
x = 4: –18 = 6B  B = –3
x = –2: –24 = – 6A  A = 4
= 4 ln(x + 2) – 3 ln(x – 4) + C
079 Error! = *
Error! = Error!  128 = A(x – 4) + Bx(x – 4) + Cx2(x – 4) + Dx3
x = 4 128 = 64 D  D = 2
x = 0 128 = –4 A  A = – 32
x = 5 128 = A + 5B + 25C + 125D = –32 + 5B + 25C + 250 = 218 + 5B + 25C
x = 6 128 = 2A + 12B + 72C + 216D = –64 + 12B + 72C +432
–90 = 5B + 25C und –240 = 12B + 72C  B = – 8 und C = – 2
* = Error! + Error! – 2 ln(x) + 2 ln(x – 4) + C = Error!
080 Error! = Error! – 4x + 14 ln(x + 4) + C
(x2 – 2) : (x + 4) = x – 4 + Error!
081 Error! = –5x e–x + Error! = – 5x e–x – 5e–x + C = C – 5e–x(x + 1)
082
Error! =
2 sin (0,5x + 3) + C
083
Error! = Error! = Error! =
Error! + C
084
Error! = Error! – Error! =
085
Error! = Error! = Error! =
Error! – Error! + C =
+C
PBZ:
Error! = Error!
–10x2 + 8x – 12 = A(2x – 3) + B x (2x – 3) + C x2
x = 0:
– 12 = – 3 A  A = 4
x = 1,5
– 22,5 = 2,25 C  C = – 10
x=2
– 36 = A + 2B + 4C = 4 + 2B – 40  B = 0
086
Error!
= Error! = 2x2 – 2 ln x – Error! + C
Error!
– 5 ln(2x – 3) –
Error!
087
088
089
090
Error! =
Error! = Error! + C = Error! + C
Error! =
u = 3x2 + 4x;du = (6x + 4) dx = Error! = Error! = 3 ln (3x2 + 4x) + C
Error! =
5x (–10)e–0,1x – Error! = – 50x e–0,1x – 500 e–0,1x + C = C – 50e–0,1x(x + 10)
Error! =
(4x2 + 2) sin(x) – Error! = (4x2 + 2) sin(x) – [–8x cos(x) – Error!] = (4x2 + 2) sin(x) + 8x cos(x) – 8
sin(x) + C = (4x2 – 6) sin(x) + 8x cos(x) + C
091
092
Error! = Error! = 2x3 – 12x2 + 93x – 372 ln(x + 4) + C
(6x3 – 3x) : (x + 4) = 6x2 – 24x + 93
–6x3 – 24x2
– 24x2 – 3x
+ 24x2 + 96x
93x
–93x – 372
Error! = Error!=
3 ln(x – 5) – 3 ln(x – 3) + Error! + C
12 = A(x – 3)2 + B (x – 3)(x – 5) + C (x – 5)
x = 3 12 = – 2 C  C = – 6
x = 5 12 = 4 A  A = 3
x=6
12 = 9A + 3B + C = 27 + 3B – 6  B = – 3
093
Error! =
3 sin(5x) + C
094
Berechnen Sie
Error!
Error! = Error! = Error! = mit u = 4x2 + 1 du = 8x dx = Error! = Error! = Error!+ C
095
Error!
= Error! = 7x2 – 3 ln x – Error! + C
096
Error! =
Error! = Error! + C = Error! + C
097
Error! =
u = 5x2 + x;du = (10x + 1) dx =
Error! = Error! = 4 ln (5x2 + x) + C
098
Error! =
5x (–5)e–0,2x – Error! = – 25x e–0,2x – 125 e–0,2x + C = C – 25e–0,2x(x + 5)
099
Error! =
(4x2 + 5) sin(x) – Error! = (4x2 + 5) sin(x) – [–8x cos(x) – Error!] = (4x2 + 5) sin(x) + 8x cos(x) – 8
sin(x) + C = (4x2 – 3) sin(x) + 8x cos(x) + C
100
Error! = Error! = Error! – 4x2 + 29x – 116 ln(x + 4) + C
(2x3 – 3x) : (x + 4) = 2x2 – 8x + 29
101
–6x3 – 8x2
– 8x2 – 3x
+ 8x2 + 32x
29x
–29x – 116
Error! = Error!=
3 ln(x – 5) – 3 ln(x – 3) + Error! + C
12 = A(x – 3)2 + B (x – 3)(x – 5) + C (x – 5)
x = 3 12 = – 2 C  C = – 6
x = 5 12 = 4 A  A = 3
x=6
12 = 9A + 3B + C = 27 + 3B – 6  B = – 3
102
Error! =
5 sin(8x) + C
103
Berechnen Sie
Error!
Error! = – Error! – Error! =
– Error! + Error! =
– Error! + Error! – Error! =
– Error! + Error! + Error! + C
104
Error! = Error! = Error! = Error! + C = 3 Error! + C
105
Error! = Error! = 2x3 – x + 2 ln(x) + Error! + C
106
Error! = Error! = Error! = ln u + C = ln(4x2 + 5) + C
107
Error!= Error! = x2 – 8x + 32 ln(x + 4) + C
108
Error! = Error! – Error! = C – 15 x cos(3x) + 5 sin(3x)
109
Error! =
– 3x2 e–x – Error! = –3x2 e–x – ( 6x e–x – Error! ) = –3x2e–x – 6xe–x – 6e–x + C =
C – 3e–x (x2 + 2x + 2)
110
Error! = Error! = Error! + 4 ln(x+3) + 3 ln(x) + C
PBZ: 7x2 + 25x + 27 = A x + B(x+3)x + C (x + 3) 2
x = 0  27 = 9C  C = 3
x = –3  15 = –3A  A = –5
x = 1  59 = A + 4B + 16C = –5 + 4B + 48  16 = 4B  B = 4
111
Error! = Error! = 2 ln(x – 5) + 3ln(x – 2) + C
x2 – 7x + 10 = 0  (x – 5)(x – 2) = 0
PBZ: 5x – 19 = A(x – 2) + B (x – 5)
x = 5  6 = 3A  A = 2
x = 2  – 9 = –3B  B = 3
112 Error! =
Division (x – 1) : (4x + 2) = Error! – Error! = Error! – Error!
= Error! = Error! – Error! + C
113.
Error! = *
Substitution u = 3 +
* = Error! =
x+8
 u–3= x+8
und
Error! = Error!
Error! = 2u – 6 ln(u) + C = 6 + 2 Error! – 6 ln(3 + Error!) + C
oder 2 x + 8 – 6 ln(3 + x + 8 ) + C weil das C die „6“ schluckt.