Evaluate the Integral integral of x(x-1)^5 with respect to x

Problem

(∫_^)(x*(x−1)5*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)*u5*d(u))

5. Distribute the u5 term across the binomial.

(∫_^)((u6+u5)*d(u))

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

(u7)/7+(u6)/6+C

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

((x−1)7)/7+((x−1)6)/6+C

Final Answer

(∫_^)(x*(x−1)5*d(x))=((x−1)7)/7+((x−1)6)/6+C

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