Evaluate the Integral

Problem

(∫_^)((ln(x))2*d(x))

Solution

1. Identify the method of integration by parts, where (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=(ln(x))2 and d(v)=d(x)

2. Differentiate u to find d(u)=2*ln(x)⋅1/x*d(x) and integrate d(v) to find v=x

3. Apply the formula for integration by parts:

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

4. Simplify the integral on the right:

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

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

6. Evaluate the inner integral:

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

(∫_^)(ln(x)*d(x))=x*ln(x)−x

7. Substitute this result back into the main equation:

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

8. Distribute the constant and simplify the final expression:

(∫_^)((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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