Răspuns:
[tex] \frac{1}{2}ln(1+\sqrt2)[/tex]
Explicație pas cu pas:
[tex]I=\int\limits^1_0 {\frac{x}{\sqrt{x^4+1}} \, dx[/tex]
Notam: [tex] x^2=t~=>~2xdx=dt~=>~xdx=\frac{1}{2}dt [/tex].
Daca x=0 => t=0.
Daca x=1 => t=1.
Integrala devine:
[tex]I=\frac{1}{2}\int\limits^1_0 {\frac{1}{\sqrt{t^2+1}} \, dt=\frac{1}{2}ln(t+\sqrt{t^2+1})|^1_0=\frac{1}{2}[ln(1+\sqrt2)-ln(0+1)]=\frac{1}{2}ln(1+\sqrt2)[/tex]
