Evaluate the Integral integral of te^(-3t) with respect to t

Problem

(∫_^)(t*e(−3*t)*d(t))

Solution

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

2. Choose the variables u=t and d(v)=e(−3*t)*d(t)

3. Differentiate u to find d(u)=d(t)

4. Integrate d(v) to find v=−1/3*e(−3*t)

5. Substitute these values into the integration by parts formula:

(∫_^)(t*e(−3*t)*d(t))=t*(−1/3*e(−3*t))−(∫_^)(−1/3*e(−3*t)*d(t))

6. Simplify the expression and the remaining integral:

(∫_^)(t*e(−3*t)*d(t))=−1/3*t*e(−3*t)+1/3*(∫_^)(e(−3*t)*d(t))

7. Evaluate the final integral:

(∫_^)(t*e(−3*t)*d(t))=−1/3*t*e(−3*t)−1/9*e(−3*t)+C

8. Factor out common terms to simplify the final result:

(∫_^)(t*e(−3*t)*d(t))=−1/9*e(−3*t)*(3*t+1)+C

Final Answer

(∫_^)(t*e(−3*t)*d(t))=−1/3*t*e(−3*t)−1/9*e(−3*t)+C

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