Evaluate the Integral

Problem

(∫_^)(√(,x)*ln(2*x)*d(x))

Solution

1. Identify the method of integration by parts, where (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

2. Assign the variables for integration by parts by letting u=ln(2*x) and d(v)=√(,x)*d(x)=x(1/2)*d(x)

3. Differentiate u to find d(u)=1/(2*x)⋅2*d(x)=1/x*d(x)

4. Integrate d(v) to find v=(x(3/2))/(3/2)=2/3*x(3/2)

5. Substitute these into the integration by parts formula:

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

6. Simplify the integral on the right:

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

7. Evaluate the remaining integral:

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

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

Final Answer

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

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