Here,
#I=intsin(ln(4x+5))dx#
Subst. #color(violet)(ln(4x+5)=u=>4x+5=e^u#
#=>4dx=e^udu=>dx=1/4e^udu#
So,
#I=intsinuxx1/4e^udu#
#=>4I=intsinue^udu...to(A)#
#"Using "color(blue)"Integration by Parts :"#
#color(red)(intf(x)g'(x)dx=f(x)g(x)-intf'(x)g(x)dx#
#=>4I=sinuinte^udu-int(cosuinte^udu)du#
#=>4I=sinu*e^u-intcosue^udu#
#=>4I=sinue^u-I_1...to(B)#
Where, #I_1=intcosue^udu#
Again #"using "color(blue)"Integration by Parts :"#
#:.I_1=cosuinte^udu-int(-sinuinte^udu)du#
#:.I_1=cosue^u+intsinue^udu+c'#
#:.I_1=cosue^u+4I+c'...to# from #(A)#
From #(B)# we get,
#:.4I=sinue^u-{cosue^u+4I}+c,where,c=-c'#
#:.4I=sinue^u-cosue^u-4I+c#
#4I+4I=sinue^u-cosue^u+c#
#8I=e^u(sinu-cosu)+c#
#I=1/8e^u(sinu-cosu)+C,where, C=c/8#
Subst, back , #color(violet)(u=ln(4x+5) and e^u=4x+5#
#I=1/8(4x+5)[sin(ln(4x+5))-cos(ln(4x+5))]+C#
#I=(4x+5)/8[sin(ln(4x+5))-cos(ln(4x+5))]+C#
