[tex]I=\int\limits^1_0 {(x^3+x^2+x+1)} \, dx= \int\limits^1_0 {x^3} \, dx +\int\limits^1_0 {x^2}\,dx+\int\limits^1_0 {x}\,dx+\int\limits^1_0 {1}\,dx\\\\I=\frac{x^4}{4}|^1_0+\frac{x^3}{3}|^1_0+\frac{x^2}{2}|^1_0+x|^1_0\\\\I=\frac{1^4}{4}-\frac{0^4}{4}+\frac{1^3}{3}-\frac{0^3}{3}+\frac{1^2}{2}-\frac{0^2}{2}+1-0\\\\I=\frac{1}{4}+\frac{1}{3}+\frac{1}{2}+1\\\\I=\frac{3+4+6+12}{12}=\frac{25}{12}[/tex]
