Evaluate the Integral integral of te^(-5t) with respect to t

Problem

(∫_^)(t*e(−5*t)*d(t))

Solution

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

2. Choose the components for the formula by letting u=t and d(v)=e(−5*t)*d(t)

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

4. Integrate d(v) to find v=−1/5*e(−5*t)

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

(∫_^)(t*e(−5*t)*d(t))=t*(−1/5*e(−5*t))−(∫_^)(−1/5*e(−5*t)*d(t))

6. Simplify the expression and the remaining integral:

(∫_^)(t*e(−5*t)*d(t))=−t/5*e(−5*t)+1/5*(∫_^)(e(−5*t)*d(t))

7. Evaluate the final integral:

(∫_^)(t*e(−5*t)*d(t))=−t/5*e(−5*t)+1/5*(−1/5*e(−5*t))+C

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

(∫_^)(t*e(−5*t)*d(t))=−1/5*t*e(−5*t)−1/25*e(−5*t)+C

Final Answer

(∫_^)(t*e(−5*t)*d(t))=−1/5*t*e(−5*t)−1/25*e(−5*t)+C

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