Evaluate the Integral

Problem

(∫_^)(ln(4*x−3)*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(4*x−3) and d(v)=d(x)

3. Differentiate u to find d(u) using the chain rule, resulting in d(u)=4/(4*x−3)*d(x)

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

5. Substitute these into the integration by parts formula:

(∫_^)(ln(4*x−3)*d(x))=x*ln(4*x−3)−(∫_^)((4*x)/(4*x−3)*d(x))

6. Rewrite the integrand of the remaining integral using algebraic manipulation:

(4*x)/(4*x−3)=(4*x−3+3)/(4*x−3)=1+3/(4*x−3)

7. Integrate the simplified expression term by term:

(∫_^)((1+3/(4*x−3))*d(x))=x+3/4*ln(4*x−3)

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

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

9. Simplify the final expression by grouping the logarithmic terms:

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

Final Answer

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

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