Evaluate the Integral integral of te^(-8t) with respect to t

Problem

(∫_^)(t*e(−8*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 variables for substitution by letting u=t and d(v)=e(−8*t)*d(t)

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

4. Integrate d(v) to find v=(∫_^)(e(−8*t)*d(t))=−1/8*e(−8*t)

5. Apply the formula for integration by parts:

t*(−1/8*e(−8*t))−(∫_^)(−1/8*e(−8*t)*d(t))

6. Simplify the expression:

−t/8*e(−8*t)+1/8*(∫_^)(e(−8*t)*d(t))

7. Evaluate the remaining integral:

−t/8*e(−8*t)+1/8*(−1/8*e(−8*t))+C

8. Factor out common terms to reach the final form:

−t/8*e(−8*t)−1/64*e(−8*t)+C

Final Answer

(∫_^)(t*e(−8*t)*d(t))=−1/8*t*e(−8*t)−1/64*e(−8*t)+C

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