Question

Sa se calculeze integrala
$\int\limits {\frac{x}{x^2+2x+2} } \, dx$

Answer 1

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: