Evaluate. (Be sure to check by differentiating!)
\[
\int\left(4 t^{3}-7\right) t^{2} d t
\]
Determine a change of variables from $t$ to $u$. Choose the correct answer below.
A. $u=4 t-7$
B. $u=t^{2}-7$
C. $u=4 t^{3}-7$
D. $u=t^{2}$
Write the integral in terms of $u$.
\[
\int\left(4 t^{3}-7\right) t^{2} d t=\int(\square) d u
\]
(Type an exact answer. Use parentheses to clearly denote the argument of each function.)
Evaluate the integral.
\[
\int\left(4 t^{3}-7\right) t^{2} d t=
\]
(Type an exact answer. Use parentheses to clearly denote the argument of each function.)

Step 1 ：Given the integral \(\int(4 t^{3}-7) t^{2} dt\), we need to evaluate it.

Step 2 ：We can integrate this polynomial term by term using the power rule, which states that the integral of \(x^n dx\) is \(\frac{1}{n+1}x^{n+1}\).

Step 3 ：Applying the power rule, we get \(\frac{2}{3}t^{6} - \frac{7}{3}t^{3}\) as the integral of \((4 t^{3}-7) t^{2}\).

Step 4 ：We also need to determine a change of variables from \(t\) to \(u\). The correct change of variable is \(u = t^{2}\).

Step 5 ：Writing the integral in terms of \(u\), we get \(\int(4u^{3/2}-7u) du\).

Step 6 ：Evaluating this integral, we again get \(\frac{2}{3}t^{6} - \frac{7}{3}t^{3}\).

Step 7 ：So, the final answer is \(\boxed{\frac{2}{3}t^{6} - \frac{7}{3}t^{3}}\).