Answer
$\displaystyle \frac{96}{35}≈2.742857$
Work Step by Step
$\displaystyle \int_0^1(1+x^2)^3dx=\int_0^1(1+x^2+x^2+x^4)(1+x^2)dx=\int_0^1(1+2x^2+x^4)(1+x^2)dx$
$\displaystyle \int_0^11+x^2+2x^2+x^2+2x^4+x^6dx=\int_0^11+3x^2+3x^4+x^6dx$
$\displaystyle (x+x^3+\frac{3x^5}{5}+\frac{x^7}{7})|_0^1$
$\displaystyle [1+1^3+\frac{3(1)^{3/5}}{5}+\frac{(1)^{7}}{7}]-[0+0^3+\frac{3(0)^{3/5}}{5}+\frac{(0)^{7}}{7}]$
$\displaystyle 2+\frac{3}{5}+\frac{1}{7}-0$
$\displaystyle 2+\frac{21}{35}+\frac{5}{35}$
$\displaystyle 2+\frac{26}{35}$
$\displaystyle \frac{70}{35}+\frac{26}{35}$
$\displaystyle \frac{96}{35}≈2.742857$
