Find the Integral ( natural log of x)^2

Problem

(∫_^)((ln(x))2*d(x))

Solution

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

2. Set the variables for the first application of integration by parts: u=(ln(x))2 and d(v)=d(x)

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

4. Substitute these into the integration by parts formula:

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

5. Simplify the integral on the right:

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

6. Apply integration by parts again for (∫_^)(ln(x)*d(x)) by setting u=ln(x) and d(v)=d(x) which gives d(u)=1/x*d(x) and v=x

7. Evaluate the inner integral:

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

(∫_^)(ln(x)*d(x))=x*ln(x)−x

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

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

9. Distribute the constant factor to reach the final form:

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

Final Answer

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

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