Integration Techniques with Examples
Transcript of Integration Techniques with Examples
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MATHEMATICS IA CALCULUS
TECHNIQUES OF INTEGRATION
WORKED EXAMPLES
Find the following integrals:
1.
3x2 − 2x + 4 dx. See worked example Page 2.
2.
1
x2
+ 1
x2 + 1
dx. See worked example Page 4.
3.
x(x + 1)2 dx. See worked example Page 5.
4.
x + 1√
xdx. See worked example Page 6.
5.
2x dx. See worked example Page 7.
6.
1
3x − 1 dx. See worked example Page 8.
7.
3sec2(5x) dx. See worked example Page 9.
8.
2√
1− 4x2 dx. See worked example Page 10.
9.
x sin x2 dx. See worked example Page 11.
10.
π4
0
sec2 x tan2 x dx. See worked example Page 12.
11.
x(2x + 1)52 dx. See worked example Page 13.
12.
4
0
√ x√
x + 1 dx. See worked example Page 14.
13.
x
2 cosh x dx. See worked example Page 16.
14.
e
1
ln x dx. See worked example Page 18.
15.
e
x cos x dx. See worked example Page 19.
16.
tan2 x dx. See worked example Page 21.
17.
cos2 x dx. See worked example Page 22.
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18.
cos4 x sin5 x dx. See worked example Page 23.
19.
cos2 x sin2 x dx. See worked example Page 24.
20.
cos4x sin5x dx. See worked example Page 25.
21.
1
(x2 + 1)3 dx, using the reduction formula for n > 1:
1
(x2 + 1)ndx =
x
(2n − 2)(x2 + 1)n−1 + 2n − 32n − 2
1
(x2 + 1)n−1 dx
See worked example Page 26.
22.
sec6 x dx, by first finding a reduction formula for
secn x dx, n ≥ 3.
See worked example Page 27.
23.
π2
0
sin12 x dx, by first finding a reduction formula for the definite integral π
2
0
sinn x dx, n ≥ 1. See worked example Page 30.
24.
5x2√
9− x2 dx. See worked example Page 33.
25.
x√
x2 − 4 dx. See worked example Page 35.
26.
1
x2 + 6x + 13
dx. See worked example Page 36.
27.
x
3 + 2x
x + 3 dx. See worked example Page 37.
28.
x
2 − 2x − 4x3 − 2x2 − 3x dx. See worked example Page 38.
29.
8x7 + 47x6 + 98x5 + 108x4 + 106x3 + 100x2 + 104x + 104
(x − 1)(x + 2)3(x2 + 2x + 2)2 dx. See worked
example Page 40.
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16/10/09 09:36 MATLAB Command Window 1 of 4
>> x=[2;3;4;5;6;7;8;9]
x =
2
3
4
56
7
8
9
>> f=(8*x.^7+47*x.^6 + 98*x.^5+108*x.^4+106*x.^3+100*x.^2+104*x+104)./((x-1).*(x+2).^3.
*(x.^2+2*x+2).^2)
f =
1307/800
305/249
985/937
394/423
728/869
972/1277
1807/2592
1240/1929
>> Ac=1./(x-1)
Ac =
1
1/2
1/3
1/4
1/5
1/6
1/7
1/8
>> Bc=1./(x+2)
Bc =
1/4
1/51/6
1/7
1/8
1/9
1/10
1/11
>> Cc=1./((x+2).^2)
Cc =
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16/10/09 09:36 MATLAB Command Window 2 of 4
1/16
1/25
1/36
1/49
1/64
1/81
1/1001/121
>> Dc=1./((x+2).^3)
Dc =
1/64
1/125
1/216
1/343
1/512
1/729
1/1000
1/1331
>> Ec=x./((x.^2+2*x+2))
Ec =
1/5
3/17
2/13
5/37
3/25
7/65
4/41
9/101
>> Fc=1./((x.^2+2*x+2))
Fc =
1/10
1/17
1/26
1/37
1/50
1/651/82
1/101
>> Gc=x./((x.^2+2*x+2).^2)
Gc =
1/50
3/289
1/169
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16/10/09 09:36 MATLAB Command Window 3 of 4
5/1369
3/1250
7/4225
2/1681
9/10201
>> Hc=1./((x.^2+2*x+2).^2)
Hc =
1/100
1/289
1/676
1/1369
1/2500
1/4225
1/6724
1/10201
>> A=[Ac,Bc,Cc,Dc,Ec,Fc,Gc,Hc,f]
A =
Columns 1 through 6
1 1/4 1/16 1/64 1/5 1/10
1/2 1/5 1/25 1/125 3/17 1/17
1/3 1/6 1/36 1/216 2/13 1/26
1/4 1/7 1/49 1/343 5/37 1/37
1/5 1/8 1/64 1/512 3/25 1/50
1/6 1/9 1/81 1/729 7/65 1/65
1/7 1/10 1/100 1/1000 4/41 1/82
1/8 1/11 1/121 1/1331 9/101 1/101
Columns 7 through 9
1/50 1/100 1307/800
3/289 1/289 305/249
1/169 1/676 985/937
5/1369 1/1369 394/423
3/1250 1/2500 728/869
7/4225 1/4225 972/1277
2/1681 1/6724 1807/2592
9/10201 1/10201 1240/1929
>> rref(A)
ans =
Columns 1 through 6
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
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16/10/09 09:36 MATLAB Command Window 4 of 4
0 0 0 0 0 1
0 0 0 0 0 0
0 0 0 0 0 0
Columns 7 through 9
0 0 1
0 0 *0 0 2
0 0 -2
0 0 7
0 0 -14
1 0 18
0 1 18