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

Answer

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Answer

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

Steps

Step 1 ：First, let's perform the u-substitution. We'll set $u = 7 - x^2$, so $du = -2x dx$. This means our integral becomes $-\int u^3 du$ 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: $\boxed{\frac{65}{4}}$

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

Answer

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