Evaluate the Integral

Problem

(∫_^)(x*ln(3+x)*d(x))

Solution

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

2. Assign variables for integration by parts by letting u=ln(3+x) and d(v)=x*d(x)

3. Differentiate u to find d(u)=1/(3+x)*d(x) and integrate d(v) to find v=(x2)/2

4. Apply the formula for integration by parts:

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

5. Simplify the integral on the right by performing polynomial long division on (x2)/(x+3)

(x2)/(x+3)=x−3+9/(x+3)

6. Substitute the result of the division back into the integral:

(∫_^)((x2)/(2*(3+x))*d(x))=1/2*(∫_^)((x−3+9/(x+3))*d(x))

7. Integrate the terms inside the parentheses:

1/2*((x2)/2−3*x+9*ln(3+x))

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

(x2)/2*ln(3+x)−(x2)/4+(3*x)/2−9/2*ln(3+x)+C

9. Factor the logarithmic terms to simplify the final expression:

(x2−9)/2*ln(3+x)−(x2)/4+(3*x)/2+C

Final Answer

(∫_^)(x*ln(3+x)*d(x))=(x2−9)/2*ln(3+x)−(x2)/4+(3*x)/2+C

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