Evaluate the Integral integral of x(x-4)^9 with respect to x

Problem

(∫_^)(x*(x−4)9*d(x))

Solution

1. Identify the substitution method to simplify the integrand. Let u=x−4

2. Differentiate the substitution equation to find d(u) Since u=x−4 then d(u)=d(x)

3. Solve for x in terms of u to substitute the remaining x term. From u=x−4 we get x=u+4

4. Substitute the expressions for x (x−4) and d(x) into the integral.

(∫_^)((u+4)*u9*d(u))

5. Distribute u9 into the binomial.

(∫_^)((u10+4*u9)*d(u))

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

(u11)/11+(4*u10)/10+C

7. Simplify the coefficients of the resulting expression.

(u11)/11+(2*u10)/5+C

8. Back-substitute u=x−4 to return to the original variable x

((x−4)11)/11+(2*(x−4)10)/5+C

Final Answer

(∫_^)(x*(x−4)9*d(x))=((x−4)11)/11+(2*(x−4)10)/5+C

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