integration techniques with examples

8/20/2019 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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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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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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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

integration techniques with examples

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