Evaluate the Integral

Problem

(∫_−3^−2)(2*x*(6−x2)3*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of the inner function 6−x2 is −2*x which is a multiple of the 2*x term in the integrand.

2. Define the substitution variable u=6−x2

3. Calculate the differential d(u) by differentiating u with respect to x

d(u)/d(x)=−2*x

d(u)=−2*x*d(x)

−d(u)=2*x*d(x)

4. Determine the new limits of integration by substituting the original x bounds into the equation for u
   Lower bound: x=−3⇒u=6−(−3)2=6−9=−3
   Upper bound: x=−2⇒u=6−(−2)2=6−4=2

5. Rewrite the integral in terms of u

(∫_−3^2)(u3*(−d(u)))

−(∫_−3^2)(u3*d(u))

6. Integrate using the power rule:

−[(u4)/4]2(−3)

7. Evaluate the definite integral by substituting the upper and lower bounds:

−(((2)4)/4−((−3)4)/4)

−(16/4−81/4)

−(−65/4)

65/4

Final Answer

(∫_−3^−2)(2*x*(6−x2)3*d(x))=65/4

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