Evaluate the given definite integral.
$\int_{- 3}^{- 2} 2 x {(7 - x^{2})}^{3} d x$
$\int_{- 3}^{- 2} 2 x {(7 - x^{2})}^{3} d x = ◻ ($ Type an integer or a simplified fraction.)

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Answer

Final Answer: $\frac{65}{4}$

Steps

Step 1 ：First, let's perform the u-substitution. We'll set $u = 7 - x^{2}$ , so $d u = - 2 x d x$ . This means our integral becomes $- \int u^{3} d u$ from $u = 7 - (- 3)^{2} = 2$ to $u = 7 - (- 2)^{2} = 3$ .

Step 2 ：Now we can find the antiderivative of $u^{3}$ , which is $(1 / 4) u^{4}$ .

Step 3 ：We'll evaluate this at 3 and 2 and subtract the two values to find the definite integral.

Step 4 ：The definite integral of the function from -3 to -2 is $65 / 4$ .

Step 5 ：Final Answer: $\frac{65}{4}$

Evaluate the given definite integral.∫−3−22x(7−x2)3dx∫−3−22x(7−x2)3dx=◻( Type an integer or a simplified fraction.)

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Evaluate the given definite integral.
$\int_{- 3}^{- 2} 2 x {(7 - x^{2})}^{3} d x$
$\int_{- 3}^{- 2} 2 x {(7 - x^{2})}^{3} d x = ◻ ($ Type an integer or a simplified fraction.)