MHB What is the meaning of Si in the integral of ln(x)cos(x)?

[leprofece]

integral ln(x).cos(x)
Here I have some clear ideas
U = lnx du = 1/x
dv = cosx so int de cosx = v = -sinx
-sinxlnx -int (sinx)/(x)

Ok I think I must integrate again
u= sinx du = cosx
dv = 1/x v = lnx
Again I got -sinxlnx -int (sinx lnx)
But I am stuck here and I don't know how to finish it??
Can you help me?

Ok i found that integral of sinx/x is Si according to a program but what does Si mean?

 

[chisigma]

integral ln(x).cos(x)
Here I have some clear ideas
U = lnx du = 1/x
dv = cosx so int de cosx = v = -sinx
-sinxlnx -int (sinx)/(x)

Ok I think I must integrate again
u= sinx du = cosx
dv = 1/x v = lnx
Again I got -sinxlnx -int (sinx lnx)
But I am stuck here and I don't know how to finish it??
Can you help me?

Ok i found that integral of sinx/x is Si according to a program but what does Si mean?

Integrating by parts You obtain...$\displaystyle \int \ln x\ \cos x\ d x = \ln x\ \sin x - \int \frac{\sin x} {x}\ d x = \ln x\ \sin x - \text{Si}\ (x) + c\ (1)$

... where...

$\displaystyle \text{Si}\ (x) = \int_{0}^{x} \frac{\sin t}{t}\ d t\ (2)$

Kind regards

$\chi$ $\sigma$