Evaluate the Integral

Problem

(∫_^)(ln(4*x)*d(x))

Solution

1. Identify the integration method. Since the integrand is a logarithmic function, use integration by parts with the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

2. Assign variables for integration by parts. Let u=ln(4*x) and d(v)=d(x)

3. Differentiate u and integrate d(v)

d(u)/d(x)=1/(4*x)⋅4=1/x

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

v=x

4. Apply the integration by parts formula.

(∫_^)(ln(4*x)*d(x))=x*ln(4*x)−(∫_^)(x⋅1/x*d(x))

5. Simplify the integral on the right side.

(∫_^)(x⋅1/x*d(x))=(∫_^)(1*d(x))

6. Evaluate the remaining integral and add the constant of integration C

(∫_^)(1*d(x))=x+C

7. Combine the terms to find the final expression.

(∫_^)(ln(4*x)*d(x))=x*ln(4*x)−x+C

Final Answer

(∫_^)(ln(4*x)*d(x))=x*ln(4*x)−x+C

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