Evaluate the Integral

Problem

(∫_−2^−1)(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 present in the integrand.

2. Define the substitution variable u=4−x2

3. Differentiate u with respect to x to find d(u)=−2*x*d(x) which implies −d(u)=2*x*d(x)

4. Change the limits of integration from x to u When x=−2 u=4−(−2)2=0 When x=−1 u=4−(−1)2=3

5. Substitute the new variables and limits into the integral.

(∫_0^3)(−u3*d(u))

6. Apply the power rule for integration, which states (∫_^)(un*d(u))=(u(n+1))/(n+1)

[−(u4)/4]30

7. Evaluate the definite integral by subtracting the value at the lower limit from the value at the upper limit.

(−3/4)−(−0/4)

8. Simplify the resulting numerical expression.

−81/4

Final Answer

(∫_−2^−1)(2*x*(4−x2)3*d(x))=−81/4

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