Evaluate the Integral

Problem

(∫_^)((ln(t))2*d(t))

Solution

1. Identify the method of integration by parts, which uses the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

2. Set u=(ln(t))2 and d(v)=d(t)

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

4. Apply the integration by parts formula:

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

5. Simplify the integral on the right:

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

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

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

7. Evaluate the remaining simple integral:

(∫_^)(ln(t)*d(t))=t*ln(t)−t

8. Substitute this result back into the expression from step 5:

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

9. Distribute the constant and simplify:

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

Final Answer

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

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