Evaluate the Integral

Problem

(∫_−3^−2)(2*x*(4−x2)3*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of the inner function 4−x2 is −2*x which is a multiple of the 2*x term in the integrand.

2. Define the substitution variable u=4−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 values into the equation for u
   Lower limit: If x=−3 then u=4−(−3)2=4−9=−5
   Upper limit: If x=−2 then u=4−(−2)2=4−4=0

5. Substitute the variables and limits into the integral:

(∫_−5^0)(−u3*d(u))

6. Evaluate the definite integral using the power rule for integration:

(∫_^)(−u3*d(u))=−(u4)/4

[−(u4)/4]0(−5)

7. Apply the Fundamental Theorem of Calculus by subtracting the value at the lower limit from the value at the upper limit:

(−0/4)−(−((−5)4)/4)

0−(−625/4)

625/4

Final Answer

(∫_−3^−2)(2*x*(4−x2)3*d(x))=625/4

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