Evaluate the Integral

Problem

(∫_4^5)(x√(,x−4)*d(x))

Solution

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

2. Determine the differential by differentiating u with respect to x which gives d(u)=d(x)

3. Express x in terms of u by rearranging the substitution equation to get x=u+4

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

5. Substitute these values into the integral to rewrite it in terms of u

(∫_0^1)((u+4)√(,u)*d(u))

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

(∫_0^1)((u(3/2)+4*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)+8/3*u(3/2)]10

8. Evaluate the definite integral by plugging in the upper limit 1 and the lower limit 0

(2/5*(1)(5/2)+8/3*(1)(3/2))−(2/5*(0)(5/2)+8/3*(0)(3/2))

9. Simplify the resulting numerical expression.

2/5+8/3

10. Find a common denominator to add the fractions.

6/15+40/15=46/15

Final Answer

(∫_4^5)(x√(,x−4)*d(x))=46/15

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