Question

integration of xlogsinx

Answer 1

this q integration is too lenthy , here is the complete solution ...

I = xlog(sinx)dx     .........1                   lim 0 t0 pi

I =(pi-x)logsin(pi-x)dx                    lim 0 to pi                      (using property of integration)
I = -xlog(sinx)dx  + pilog(sinx)dx ................2              lim 0 to pi
adding 1 & 2
2I = pilogsinxdx or
2I/pi = logsinxdx  .....................3           lim 0 to pi    

2Ipi = logsinxdx + logsinpi-x  dx      lim 0 to pi/2                (by property)
2I/PI = 2logsinx dx ................4      lim 0 to pi/2
2I/pi = 2logcosx dx ................5   lim 0 to pi/2           (by property)

adding 4 & 5

4I/pi =2log(sin2x)/2 dx       lim 0 to pi/2
2I/pi = logsin2xdx - log2 dx        from 0 to pi/2    
2I/pi =   I1 - log2dx                      lim 0 to pi

from this eq 6 integrating logsin2x seperately

I1 = logsin2xdx             lim 0 to pi/2
put 2x =t
I1 = (logsintdt)/2  lim 0 to pi

now interchanging variable t with x

I1 = logsinxdx/2            lim 0 to pi ........................7
from eq 3 & 7
I1 = I/pi .........8
putting eq 8 in 6

then

2I/pi = I/Pi  - log2dx    lim 0 to pi
I/pi = -(log2)x      lim 0 to pi
I/pi   =-pilog2

:-):-):-):-):-):-)
MERRY CHRISTMAS