Evaluate the Integral

Problem

(∫_^)(x/√(,2*x−1)*d(x))

Solution

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

2. Differentiate the substitution to find the relationship between d(u) and d(x)

d(u)/d(x)=2

d(x)=1/2*d(u)

3. Solve for x in terms of u to substitute the numerator.

u=2*x−1

2*x=u+1

x=(u+1)/2

4. Substitute all components into the integral.

(∫_^)((u+1)/2/√(,u)1/2*d(u))

5. Simplify the constant factors and the integrand.

1/4*(∫_^)((u+1)/(u(1/2))*d(u))

1/4*(∫_^)((u(1/2)+u(−1/2))*d(u))

6. Integrate each term using the power rule.

1/4*(2/3*u(3/2)+2*u(1/2))+C

7. Distribute the constant factor.

1/6*u(3/2)+1/2*u(1/2)+C

8. Back-substitute u=2*x−1 to return to the original variable.

1/6*(2*x−1)(3/2)+1/2*(2*x−1)(1/2)+C

Final Answer

(∫_^)(x/√(,2*x−1)*d(x))=1/6*(2*x−1)(3/2)+1/2*(2*x−1)(1/2)+C

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