We need
#intx^ndx=x^(n+1)/(n+1)+C (n!=-1)#
#sin^2x+cos^2x=1#
Rewrite the integral (we apply linearity)
#int((4x+3)dx)/sqrt(1-x^2)=int(4xdx)/sqrt(1-x^2)+int(3dx)/sqrt(1-x^2)#
#=4int(xdx)/sqrt(1-x^2)+3intdx/sqrt(1-x^2)#
Let #u=1-x^2#, #=>#, #du=-2xdx#
#int(xdx)/sqrt(1-x^2)=-1/2int(du)/sqrtu#
#=-1/2sqrtu/(1/2)=-sqrtu=-sqrt(1-x^2)#
Let #x=sin theta#, #=>#, #dx=costheta(d theta)#
#sqrt(1-x^2)=sqrt(1-sin^2theta)=sqrt (cos^2theta)=costheta#
#int(dx)/sqrt(1-x^2)=int(costheta d theta)/costheta=intd theta=theta#
#=arcsinx#
Putting it all together
#int((4x+3)dx)/sqrt(1-x^2)=-4sqrt(1-x^2)+3arcsinx+C#
