Evaluate the Integral

Problem

(∫_3^4)(x√(,x−3)*d(x))

Solution

1. Identify the substitution to simplify the integrand. Let u=x−3

2. Differentiate the substitution to find the relationship between d(x) and d(u) Since u=x−3 then d(u)=d(x)

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

4. Change the limits of integration. When x=3 u=3−3=0 When x=4 u=4−3=1

5. Substitute all components into the integral.

(∫_0^1)((u+3)√(,u)*d(u))

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

(∫_0^1)((u(3/2)+3*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)+3⋅2/3*u(3/2)]10

8. Simplify the expression before evaluating.

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

9. Evaluate at the upper and lower limits.

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

10. Calculate the final numerical value.

2/5+2=2/5+10/5=12/5

Final Answer

(∫_3^4)(x√(,x−3)*d(x))=12/5

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