Chapter 6.3, Problem 49E

Calculus Let $B=\left\{1,x,x^2,x^3\right\}$ be a basis for $P_3$, and $T:P_3\to P_4$ be the linear transformation represented by $T\left(x^k\right)={\displaystyle {\int }_0^x{t}^{k}dt}$.

$\left(a\right)$ Find the matrix $A$ for $T$ with respect to $B$ and the standard basis for $P_4$.

$\left(b\right)$ Use $A$ to integrate $p\left(x\right)=8-4x+3x^3$.