Evaluate the Integral integral of x(x+1)^8 with respect to x

Problem

(∫_^)(x*(x+1)8*d(x))

Solution

1. Identify the substitution method to simplify the integrand. Let u=x+1

2. Differentiate the substitution equation to find d(u)=d(x)

3. Solve for x in terms of u to substitute the remaining x term, giving x=u−1

4. Substitute these expressions into the integral to rewrite it in terms of u

(∫_^)((u−1)*u8*d(u))

5. Distribute the u8 term across the parentheses.

(∫_^)((u9−u8)*d(u))

6. Integrate each term using the power rule (∫_^)(un*d(u))=(u(n+1))/(n+1)

(u10)/10−(u9)/9+C

7. Back-substitute u=x+1 to return to the original variable x

((x+1)10)/10−((x+1)9)/9+C

Final Answer

(∫_^)(x*(x+1)8*d(x))=((x+1)10)/10−((x+1)9)/9+C

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