Evaluate the Integral

Problem

(∫_^)(ln(5*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 the integration by parts formula by letting u=ln(5*x) and d(v)=d(x)

3. Differentiate u to find d(u)=1/(5*x)⋅5*d(x)=1/x*d(x)

4. Integrate d(v) to find v=x

5. Substitute these values into the integration by parts formula.

(∫_^)(ln(5*x)*d(x))=x*ln(5*x)−(∫_^)(x⋅1/x*d(x))

6. Simplify the integrand in the second term.

(∫_^)(ln(5*x)*d(x))=x*ln(5*x)−(∫_^)(1*d(x))

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

(∫_^)(ln(5*x)*d(x))=x*ln(5*x)−x+C

Final Answer

(∫_^)(ln(5*x)*d(x))=x*ln(5*x)−x+C

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