Evaluate the Integral

Problem

(∫_^)(14*x*ln(3*x)*d(x))

Solution

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

2. Choose the parts by letting u=ln(3*x) and d(v)=14*x*d(x)

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

4. Integrate d(v) to find v=(∫_^)(14*x*d(x))=7*x2

5. Apply the formula for integration by parts:

(∫_^)(14*x*ln(3*x)*d(x))=(7*x2)*(ln(3*x))−(∫_^)(7*x2⋅1/x*d(x))

6. Simplify the integral on the right:

(∫_^)(7*x2⋅1/x*d(x))=(∫_^)(7*x*d(x))

7. Evaluate the remaining integral:

(∫_^)(7*x*d(x))=(7*x2)/2

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

Final Answer

(∫_^)(14*x*ln(3*x)*d(x))=7*x2*ln(3*x)−(7*x2)/2+C

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