Let $p(1)=-2, p^{\prime}(1)=-2, p(4)=-2, p^{\prime}(4)=8$.
Evaluate the integral.
\[
\int_{1}^{4} t p^{\prime \prime}(t) d t=
\]

Answer

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Answer

Evaluate the integral: $\int_{1}^{4} t p''(t) dt = \boxed{34}$

Steps

Step 1 ：Assume $p(t) = at^3 + bt^2 + ct + d$ is a cubic polynomial

Step 2 ：Use given conditions to form a system of equations: $\begin{cases} a+b+c+d=-2\\ 3a+2b+c=-2\\ 64a+16b+4c+d=-2\\ 48a+8b+c=8\end{cases}$

Step 3 ：Solve the system of equations to find $a=\frac{2}{3}$, $b=-\frac{10}{3}$, $c=\frac{8}{3}$, and $d=-2$

Step 4 ：Find the second derivative: $p''(t) = 4t - \frac{20}{3}$

Step 5 ：Evaluate the integral: $\int_{1}^{4} t p''(t) dt = \boxed{34}$