Evaluate the Integral

Problem

(∫_−4^2)(2*x+4*d(x))

Solution

1. Identify the antiderivative of the integrand 2*x+4 using the power rule for integration.

2. Apply the power rule to each term, where (∫_^)(xn*d(x))=(x(n+1))/(n+1)

3. Find the antiderivative F(x) by calculating (∫_^)(2*x*d(x))+(∫_^)(4*d(x))

F(x)=x2+4*x

4. Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper limit 2 and the lower limit −4

(∫_−4^2)(2*x+4*d(x))=[x2+4*x]2(−4)

5. Substitute the upper limit into the expression.

(2)2+4*(2)=4+8=12

6. Substitute the lower limit into the expression.

(−4)2+4*(−4)=16−16=0

7. Subtract the lower limit value from the upper limit value.

12−0=12

Final Answer

(∫_−4^2)(2*x+4*d(x))=12

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