Evaluate the Integral

Problem

(∫_^)((−x8+4)6*x7*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of the inner function −x8+4 is a multiple of the remaining factor x7

2. Define the substitution variable u=−x8+4

3. Differentiate u with respect to x to find d(u)=−8*x7*d(x)

4. Rearrange the differential to solve for the terms present in the integral: −1/8*d(u)=x7*d(x)

5. Substitute the variables into the integral to rewrite it in terms of u

(∫_^)(u6*(−1/8)*d(u))

6. Factor out the constant coefficient:

−1/8*(∫_^)(u6*d(u))

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

−1/8*((u7)/7)+C

8. Simplify the expression by multiplying the denominators:

−(u7)/56+C

9. Back-substitute the original expression for u to get the final result in terms of x

−((−x8+4)7)/56+C

Final Answer

(∫_^)((−x8+4)6*x7*d(x))=−((−x8+4)7)/56+C

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