Evaluate the Integral

Problem

(∫_1^5)(x/√(,2*x−1)*d(x))

Solution

1. Identify the substitution to simplify the integrand. Let u=√(,2*x−1)

2. Rearrange the substitution to solve for x Squaring both sides gives u2=2*x−1 which implies x=(u2+1)/2

3. Differentiate x with respect to u to find the differential d(x)

d(x)/d(u)=u

d(x)=u*d(u)

4. Change the limits of integration. When x=1 u=√(,2*(1)−1)=1 When x=5 u=√(,2*(5)−1)=3

5. Substitute the expressions for x √(,2*x−1) and d(x) into the integral.

(∫_1^3)((u2+1)/2/u⋅u*d(u))

6. Simplify the integrand by canceling u

(∫_1^3)((u2+1)/2*d(u))

7. Factor out the constant and integrate the polynomial.

1/2*(∫_1^3)((u2+1)*d(u))

1/2*[(u3)/3+u]31

8. Evaluate the definite integral at the boundaries.

1/2*[(3/3+3)−(1/3+1)]

1/2*[(9+3)−(1/3+1)]

1/2*[12−4/3]

1/2*[36/3−4/3]

1/2⋅32/3

16/3

Final Answer

(∫_1^5)(x/√(,2*x−1)*d(x))=16/3

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