Evaluate the Integral

Problem

(∫_2^3)(x√(,x−2)*d(x))

Solution

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

2. Calculate the differential d(u) and express x in terms of u Since u=x−2 then d(u)=d(x) and x=u+2

3. Determine the new limits of integration. When x=2 u=2−2=0 When x=3 u=3−2=1

4. Substitute the expressions into the integral.

(∫_0^1)((u+2)√(,u)*d(u))

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

(∫_0^1)((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)

[2/5*u(5/2)+4/3*u(3/2)]10

7. Evaluate the definite integral at the upper and lower limits.

(2/5*(1)(5/2)+4/3*(1)(3/2))−(2/5*(0)(5/2)+4/3*(0)(3/2))

8. Simplify the resulting fractions.

2/5+4/3=6/15+20/15=26/15

Final Answer

(∫_2^3)(x√(,x−2)*d(x))=26/15

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