Evaluate the Integral

Problem

(∫_^)(√(,x−1)*d(x))

Solution

1. Identify the integral as a power function involving a linear substitution.

2. Substitute u=x−1 which implies d(u)=d(x)

3. Rewrite the integral in terms of u

(∫_^)(u(1/2)*d(u))

4. Apply the power rule for integration, which states (∫_^)(un*d(u))=(u(n+1))/(n+1)+C

(u(3/2))/(3/2)+C

5. Simplify the coefficient:

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

6. Back-substitute x−1 for u to get the final result.

Final Answer

(∫_^)(√(,x−1)*d(x))=2/3*(x−1)(3/2)+C

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