Evaluate the Integral

Problem

(∫_5^6)(x√(,x−5)*d(x))

Solution

1. Identify the substitution to simplify the integrand. Let u=x−5

2. Differentiate the substitution to find d(u) Since u=x−5 then d(u)=d(x)

3. Express x in terms of u From u=x−5 we get x=u+5

4. Change the limits of integration. When x=5 u=5−5=0 When x=6 u=6−5=1

5. Substitute the expressions into the integral.

(∫_0^1)((u+5)√(,u)*d(u))

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

(∫_0^1)((u(3/2)+5*u(1/2))*d(u))

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

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

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

(2/5*(1)(5/2)+10/3*(1)(3/2))−(2/5*(0)(5/2)+10/3*(0)(3/2))

9. Simplify the numerical expression.

2/5+10/3

10. Find a common denominator to add the fractions.

6/15+50/15=56/15

Final Answer

(∫_5^6)(x√(,x−5)*d(x))=56/15

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