Evaluate the Integral

Problem

(∫_^)(√(,x)*ln(x)*d(x))

Solution

1. Identify the integration method. Since the integrand is a product of an algebraic function √(,x) and a logarithmic function ln(x) use integration by parts.

2. Choose the parts for integration by parts using the LIATE rule. Let u=ln(x) and d(v)=√(,x)*d(x)=x(1/2)*d(x)

3. Differentiate u and integrate d(v) to find d(u) and v

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

v=(∫_^)(x(1/2)*d(x))=2/3*x(3/2)

4. Apply the integration by parts formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

(∫_^)(√(,x)*ln(x)*d(x))=2/3*x(3/2)*ln(x)−(∫_^)(2/3*x(3/2)⋅1/x*d(x))

5. Simplify the integral on the right side.

(∫_^)(2/3*x(3/2)⋅1/x*d(x))=2/3*(∫_^)(x(1/2)*d(x))

6. Evaluate the remaining integral.

2/3*(∫_^)(x(1/2)*d(x))=2/3⋅2/3*x(3/2)=4/9*x(3/2)

7. Combine the terms and add the constant of integration C

2/3*x(3/2)*ln(x)−4/9*x(3/2)+C

Final Answer

(∫_^)(√(,x)*ln(x)*d(x))=2/3*x(3/2)*ln(x)−4/9*x(3/2)+C

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