Evaluate the Integral

Problem

(∫_^)(x√(,x−7)*d(x))

Solution

1. Identify the substitution to simplify the integrand by letting u=x−7

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

3. Solve for x in terms of u to get x=u+7

4. Substitute these expressions into the integral to change the variable from x to u

(∫_^)((u+7)√(,u)*d(u))

5. Distribute the √(,u) (which is u(1/2) into the parentheses.

(∫_^)((u(3/2)+7*u(1/2))*d(u))

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

(u(5/2))/(5/2)+(7*u(3/2))/(3/2)+C

7. Simplify the coefficients of the resulting terms.

2/5*u(5/2)+14/3*u(3/2)+C

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

2/5*(x−7)(5/2)+14/3*(x−7)(3/2)+C

Final Answer

(∫_^)(x√(,x−7)*d(x))=2/5*(x−7)(5/2)+14/3*(x−7)(3/2)+C

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