Evaluate the Integral

Problem

(∫_^)(x√(,x+2)*d(x))

Solution

1. Identify the substitution to simplify the integrand. Let u=x+2

2. Differentiate the substitution to find the relationship between d(x) and d(u)

u=x+2

d(u)/d(x)=1

d(u)=d(x)

3. Solve for x in terms of u to substitute the remaining x in the integrand.

x=u−2

4. Substitute the expressions for x √(,x+2) and d(x) into the integral.

(∫_^)((u−2)√(,u)*d(u))

5. Distribute the √(,u) (which is u(1/2) across the terms in the parentheses.

(∫_^)((u(3/2)−2*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)−2(u(3/2))/(3/2)+C

7. Simplify the coefficients.

2/5*u(5/2)−4/3*u(3/2)+C

8. Back-substitute u=x+2 to express the final answer in terms of x

2/5*(x+2)(5/2)−4/3*(x+2)(3/2)+C

Final Answer

(∫_^)(x√(,x+2)*d(x))=2/5*(x+2)(5/2)−4/3*(x+2)(3/2)+C

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