Evaluate. (Be sure to check by differentiating!)
$\int (4 t^{3} - 7) t^{2} d t$
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 $u$ .
$\int (4 t^{3} - 7) t^{2} d t = \int (◻) d u$
(Type an exact answer. Use parentheses to clearly denote the argument of each function.)
Evaluate the integral.
$\int (4 t^{3} - 7) t^{2} d t =$
(Type an exact answer. Use parentheses to clearly denote the argument of each function.)

Answer

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Answer

So, the final answer is $\frac{2}{3} t^{6} - \frac{7}{3} t^{3}$ .

Steps

Step 1 ：Given the integral $\int (4 t^{3} - 7) t^{2} d t$ , 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 $x^{n} d x$ 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 $\int (4 u^{3 / 2} - 7 u) d u$ .

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 $\frac{2}{3} t^{6} - \frac{7}{3} t^{3}$ .

Answer

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