Evaluate the Integral

Problem

(∫_^)((x−1)√(,x)*d(x))

Solution

1. Rewrite the integrand by expressing the square root as a fractional power.

√(,x)=x(1/2)

2. Distribute the x(1/2) term into the parentheses.

(x−1)*x(1/2)=x1⋅x(1/2)−1⋅x(1/2)

3. Simplify the exponents using the product rule for powers.

x(3/2)−x(1/2)

4. Apply the power rule for integration, (∫_^)(xn*d(x))=(x(n+1))/(n+1)+C to each term.

(∫_^)(x(3/2)*d(x))−(∫_^)(x(1/2)*d(x))

5. Calculate the resulting coefficients and add the constant of integration.

(x(5/2))/(5/2)−(x(3/2))/(3/2)+C

6. Simplify the fractions by multiplying by the reciprocals.

2/5*x(5/2)−2/3*x(3/2)+C

Final Answer

(∫_^)((x−1)√(,x)*d(x))=2/5*x(5/2)−2/3*x(3/2)+C

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