Evaluate the Integral

Problem

(∫_^)(3*x5−8*x3+2*d(x))

Solution

1. Identify the integral as a sum of power functions, allowing the use of the sum rule for integration.

2. Apply the power rule for integration, which states that (∫_^)(xn*d(x))=(x(n+1))/(n+1) for n≠−1

3. Integrate the first term 3*x5

(∫_^)(3*x5*d(x))=3⋅(x6)/6=1/2*x6

4. Integrate the second term −8*x3

(∫_^)(−8*x3*d(x))=−8⋅(x4)/4=−2*x4

5. Integrate the constant term 2

(∫_^)(2*d(x))=2*x

6. Combine the results and add the constant of integration C

Final Answer

(∫_^)(3*x5−8*x3+2*d(x))=1/2*x6−2*x4+2*x+C

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