Evaluate $\int_0^1 x\left(x^2 + 1\right)^3 dx$.

Explanation

1. 01

   Put $t = x^2 + 1$, $dt = 2x\,dx$; $x = 0 \to t = 1$, $x = 1 \to t = 2$.

2. 02

   $\frac{1}{2}\int_1^2 t^3\,dt = \frac{1}{2} \cdot \frac{16 - 1}{4} = \frac{15}{8}$.