Răspuns
∫xdx/(x²+2x+2)=1/2∫2xdx/(x²+2x+2)=1/2∫(2x+2)dx/(x²+2x+2)-1/2∫2dx/(x²+2x+2)
F1(x)=1/2∫(2x+2)dx/(x²+2x+2)
x²+2x+2=t
2x+2)dx=dt
F1(t)=1/2∫dt/t=1/2*lnt +c1
Revii la x
F1(x)=1/2 *ln(x²+2x+2)+c1
F2(x)=∫dx/(x²+2x+2)=∫dx/[(x²+2x+1)+1]
∫dx/(x+1)²+1
x+1=y dx=dy
F2(y)=∫dy/(y²+1)=arctgy+c2
Revii la x
F2(x)=arctg(x+1)+c2
F(x)=F1(x)-F2(x)=
1/2ln(x²+2x+2)+c1+arctg(x+1)+c2=
1/2ln(x²+2x+2)+arctg(x+1)+C
Explicație pas cu pas:
