Evaluate the Integral

Problem

(∫_^)(ln(x)/√(,x)*d(x))

Solution

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

2. Assign variables for integration by parts using the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=ln(x) and d(v)=x(−1/2)*d(x)

3. Differentiate u to find d(u)=1/x*d(x)

4. Integrate d(v) to find v=(∫_^)(x(−1/2)*d(x))=2*x(1/2)

5. Substitute these into the integration by parts formula.

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

6. Simplify the integral on the right side.

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

7. Evaluate the remaining integral.

(∫_^)(2*x(−1/2)*d(x))=4*x(1/2)

8. Combine the results and add the constant of integration C

2√(,x)*ln(x)−4√(,x)+C

Final Answer

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

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