по частям
u=arctgx ⇒ du=dx/(1+x^2)
dv=(x+1)dx ⇒ v=(x^2/2)+x
u*v - ∫ v*du=
=((x^2/2)+x)*arctgx - ∫ ((x^2/2)+x)dx/(x^2+1)=
=((x^2/2)+x)*arctgx - (1/2)∫ (x^2+2x)dx/(x^2+1)=
=((x^2/2)+x)*arctgx - (1/2)∫ (x^2+1 +2x-1)dx/(x^2+1)=
=((x^2/2)+x)*arctgx - (1/2)∫ (1 + (2x-1)/(x^2+1))dx=
=((x^2/2)+x)*arctgx - (1/2)x - (1/2) ln(x^2+1) +(1/2)arctgx +C
19б)
по частям
u= x^2 ⇒ du=2xdx
dv=x*e^(-x^2)dx ⇒ v= ∫ x*e^(-x^2)dx= ( замена (-x^2=t; dt=-2xdx)=
=(-1/2) ∫e^(t)dt=(-1/2)e^(t)=(-1/2)e^(-x^2)
u*v - ∫ v*du=
=(-1/2)*x^2*e^(-x^2) - ∫ (-1/2) * e^(-x^2) *2xdx=
=(-1/2)*x^2*e^(-x^2) - (1/2) ∫ * e^(-x^2) *(-2x)dx=
=(-1/2)*x^2*e^(-x^2) - (1/2) * e^(-x^2) + C