Evaluate the Integral

Problem

(∫_^)(3*x^5−8*x^3+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 (∫_^)(x^n*d(x))=(x^(n+1))/(n+1) for n≠−1

3. Integrate the first term 3*x^5

(∫_^)(3*x^5*d(x))=3⋅(x^6)/6=1/2*x^6

4. Integrate the second term −8*x^3

(∫_^)(−8*x^3*d(x))=−8⋅(x^4)/4=−2*x^4

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*x^5−8*x^3+2*d(x))=1/2*x^6−2*x^4+2*x+C

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