Evaluate the Integral

Problem

(∫_−1^1)((4*x2−1)*(x+3)*d(x))

Solution

1. Expand the integrand by multiplying the binomials using the FOIL method.

(4*x2−1)*(x+3)=4*x3+12*x2−x−3

2. Rewrite the definite integral with the expanded polynomial.

(∫_−1^1)((4*x3+12*x2−x−3)*d(x))

3. Apply the power rule for integration, which states (∫_^)(xn*d(x))=(x(n+1))/(n+1) to find the antiderivative.

(∫_^)((4*x3+12*x2−x−3)*d(x))=x4+4*x3−(x2)/2−3*x

4. Evaluate the antiderivative at the upper limit of integration (x=1.

(1)4+4*(1)3−((1)2)/2−3*(1)=1+4−0.5−3

=1.5

5. Evaluate the antiderivative at the lower limit of integration (x=−1.

(−1)4+4*(−1)3−((−1)2)/2−3*(−1)=1−4−0.5+3

=−0.5

6. Subtract the value at the lower limit from the value at the upper limit according to the Fundamental Theorem of Calculus.

1.5−(−0.5)=2

Final Answer

(∫_−1^1)((4*x2−1)*(x+3)*d(x))=2

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