Evaluate the Integral

Problem

(∫_^)((−x^8+4)^6*x^7*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of the inner function −x^8+4 is a multiple of the remaining factor x^7

2. Define the substitution variable u=−x^8+4

3. Differentiate u with respect to x to find d(u)=−8*x^7*d(x)

4. Rearrange the differential to solve for the terms present in the integral: −1/8*d(u)=x^7*d(x)

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

(∫_^)(u^6*(−1/8)*d(u))

6. Factor out the constant coefficient:

−1/8*(∫_^)(u^6*d(u))

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

−1/8*((u^7)/7)+C

8. Simplify the expression by multiplying the denominators:

−(u^7)/56+C

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

−((−x^8+4)^7)/56+C

Final Answer

(∫_^)((−x^8+4)^6*x^7*d(x))=−((−x^8+4)^7)/56+C

Want more problems? Check here!