Evaluate the Integral

Problem

(∫_^)(ln(2*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(2*x+1) and d(v)=d(x)

3. Differentiate u to find d(u) using the chain rule, resulting in d(u)=2/(2*x+1)*d(x)

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

5. Substitute these into the integration by parts formula.

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

6. Rewrite the integrand of the remaining integral by adding and subtracting 1 in the numerator.

(2*x)/(2*x+1)=(2*x+1−1)/(2*x+1)

7. Simplify the fraction into two separate terms.

(2*x+1−1)/(2*x+1)=1−1/(2*x+1)

8. Integrate the simplified terms individually.

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

9. Combine all parts and include the constant of integration C

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

10. Factor the natural log terms to simplify the final expression.

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

Final Answer

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

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