Evaluate the Integral

Problem

(∫_^)((2*x−1)*ln(7*x)*d(x))

Solution

1. Identify the integration by parts formula, which is (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

2. Choose the components for substitution by letting u=ln(7*x) and d(v)=(2*x−1)*d(x)

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

4. Integrate d(v) to find v=x2−x

5. Apply the integration by parts formula:

(∫_^)((2*x−1)*ln(7*x)*d(x))=(x2−x)*ln(7*x)−(∫_^)((x2−x)1/x*d(x))

6. Simplify the integrand in the second term:

(∫_^)((x2−x)1/x*d(x))=(∫_^)((x−1)*d(x))

7. Evaluate the remaining integral:

(∫_^)((x−1)*d(x))=(x2)/2−x

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

(∫_^)((2*x−1)*ln(7*x)*d(x))=(x2−x)*ln(7*x)−((x2)/2−x)+C

9. Distribute the negative sign to reach the final form:

(∫_^)((2*x−1)*ln(7*x)*d(x))=(x2−x)*ln(7*x)−(x2)/2+x+C

Final Answer

(∫_^)((2*x−1)*ln(7*x)*d(x))=(x2−x)*ln(7*x)−(x2)/2+x+C

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