Evaluate the Integral

Problem

(∫_^)(ln(4*x+1)*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+1) and d(v)=d(x)

3. Differentiate u to find d(u)=4/(4*x+1)*d(x) and integrate d(v) to find v=x

4. Apply the formula for integration by parts:

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

5. Rewrite the integrand of the remaining integral using algebraic manipulation by adding and subtracting 1 in the numerator:

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

6. Integrate the simplified terms:

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

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

x*ln(4*x+1)−(x−1/4*ln(4*x+1))+C

8. Simplify the expression by grouping the natural log terms:

(x+1/4)*ln(4*x+1)−x+C

Final Answer

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

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