Here,
#I=2piint_0^1 xcos^-1x#
Subst. #cos^-1x=u=>x=cosu=>dx=-sinudu#
#x=0=>cosu=0=>u=pi/2and#
# x=1=>cosu=1=>u=0#
So,
#I=2piint_(pi/2)^0 cosu*u(-sinu)du#
#=2piint_0^(pi/2) usinucosudu.to[becauseint_a^bf(x)dx=-int_b^af(x)dx]#
#=piint_0^(pi/2)usin2udu...to[becausesin2theta=2sinthetacostheta]#
#"Using "color(red)"Integratio by Parts :"#
#color(blue)(intu*vdx=u*intvdx-int(u'intvdx)dx#
Take, #u=u andv=sin2u=>u'=1 and intvdu=-(cos2u)/2#
#I=pi{[u*(-cos2u)/2]_0^(pi/2)-int_0^(pi/2)(-cos2u)/2du}#
#=pi{[pi/2*(-cospi)/2-0]+1/2[(sin2u)/2]_0^(pi/2)}#
#=pi{pi/2(1/2)+1/4[sinpi-sin0]}#
#=pi{pi/4}#
#=pi^2/4#