Numerical Integration - Numerical Methods in Engineering with MATLAB - page 229

Solution As the integrand issmooth and free of singularities, wecoulduse Gauss-
Legendrequadrature. However, the exact integralcan obtainedwith the Gauss-
Chebyshev formula. We write
1

1

x 2 2

1

1

x 2 3 / 2 dx

−

− = √ 1

x 2 dx − − 1 − 1

x 2 ) 2 is apolynomialofdegree four,sothatGauss-
Chebyshev quadrature isexact with three nodes.
The abscissas of the nodes areobtained fromEq. (6.32).Substituting n

The numerator f ( x ) = (1 −

= 3 , we

get

(2 i − 1) π

x i = cos , i = 1 , 2 , 3

2(3)

Therefore,

√ 3
2

cos 6 =

x 1 =

cos 2 =

x 2 = 0

√ 3
2

cos 5

6 =

x 2 =

and Eq. (6.31) yields
1

1

x 2 3 / 2 dx

3

1

x i 2

3

− = −

− 1

i = 1

1

2 2 1

3 3

4

3
4

3 8

0) 2 = − + (1 − + − =

EXAMPLE 6.9
Use Gaussian integration to evaluate 0 . 5
0

π

cos x ln x dx .

Solution Wesplit the integral into two parts:
0 . 5

1 1

cos π x ln x dx = cos π x ln x dx − cos π x ln x dx

0 0 0 . 5

The first integralon the right-hand side, which contains a logarithmicsingularityat
x = 0, can becomputedwith the specialGaussian quadrature in Eq. (6.38).Choosing

n = 4, wehave

1

4

cos π x ln x dx ≈ − A i cos π x i

0

i = 1