Numerical Integration - Numerical Methods in Engineering with MATLAB - page 230

where x i and A i are giveninTable 6.6. The sum isevaluatedinthe following table:

x i cos π x i A i A i cos π x i

0 . 041 448 0 . 991 5340 . 383 464 0 . 380 218

0 . 245 275 0 . 717 525 0 . 386 875 0 . 277 592

0 . 556 165 − 0 . 175 533 0 . 190435 − 0 . 033 428

0 . 848 982 − 0 . 889 5500 . 039 225 − 0 . 034 892

= 0 . 589 490

Thus

1

cos π x ln x dx ≈ − 0 . 589 490

0

The second integral isfree of singularities, so that itcan beevaluatedwith Gauss-
Legendrequadrature.Choosing again n

= 4, wehave

1

4

cos π x ln x dx 0 . 25 A i cos π x i ln x i ≈

0 . 5

i = 1

where the nodal abscissas are (see Eq. (6.28))

1 + 0 . 5 1 − 0 . 5

x i = + ξ i = 0 . 75 + 0 . 25 ξ i

2 2

Looking up ξ i and A i in Table 6.3 leads to the following computations:

ξ i x i cos π x i ln x i A i A i cos π x i ln x i

− 0 . 861 136 0 . 534 716 0 . 068 141 0 . 347 855 0 . 023 703

− 0 . 339 981 0 . 665 005 0 . 202 133 0 . 652 145 0 . 131 820

0 . 339 981 0 . 834 995 0 . 156 638 0 . 652 145 0 . 102 151

0 . 861 136 0 . 965 284 0 . 035 123 0 . 347 855 0 . 012 218

= 0 . 269 892

fromwhich

1

π . . = .

cos x ln x dx ≈ 0 25(0 269 892) 0 067 473

0 . 5

Therefore,
1

cos π x ln x dx ≈ − 0 . 589 490 − 0 . 067 473 =− 0 . 656 96 3

0

which iscorrect to six decimal places.