1) ∫xarctgx= [u= arctgx; du=1/(1+x2); dv= xdx; v=x2/2]= x2/2arctgx-1/2∫x2dx/(1+x2)= x2/2arctgx-x/2+1/2arctgx+C= (x2+1)/2arctgx-x/2+C
S = [0;√3] ∫xarctgx
S= 2arctg√3-√3/2-0-0= 2π/3-√3/2
3) S= [0;2π]∫(2cosx-cosx)dx= 4*[0;π/2]∫(2cosx-cosx)dx=4*[0;π/2]∫cosxdx= 4sin(π/2)-sin0=4
1) = [u= arctg√(3x-1); du=dx/x(6x-2); dv= dx; v=x]= x arctg√(3x-1)-∫dx/(6x-2)= x arctgx-1/6∫d(6x-2)/(6x-2)= x arctgx-1/6ln(6x-2)+C
2) = ∫(x3+x2)dx+ ∫dx/(x(x-1))= x4/4+x3/3-∫dx/x+∫d(x-1)/(x-1)= x4/4+x3/3-lnx+ln(x-1)+C
3) = ∫(3/(x-1)+1/(x+1)3)dx= 3ln(x-1)-1/(2(x+1)2)+C
4) =4∫dx/(x+3)2+4∫(x-1)dx/((x+1)2+1)= [t= x+1; x= t-1; dx= dt]= 4∫d(x+3)/(x+3)2+2∫dt2/(t2+1)-4∫dt/(t2+1)= -4/(x+3)+2ln(x2+2x+2)-arctg(x+1)+C