Evaluate the Integral integral of te^(-6t) with respect to t

Problem

(∫_^)(t*e(−6*t)*d(t))

Solution

1. Identify the integration method. Since the integrand is a product of an algebraic term t and an exponential term e(−6*t) use integration by parts.

2. Assign variables for the integration by parts formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=t and d(v)=e(−6*t)*d(t)

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

4. Integrate d(v) to find v=(∫_^)(e(−6*t)*d(t))=−1/6*e(−6*t)

5. Apply the formula for integration by parts.

(∫_^)(t*e(−6*t)*d(t))=t*(−1/6*e(−6*t))−(∫_^)(−1/6*e(−6*t)*d(t))

6. Simplify the expression and the remaining integral.

(∫_^)(t*e(−6*t)*d(t))=−t/6*e(−6*t)+1/6*(∫_^)(e(−6*t)*d(t))

7. Evaluate the final integral.

(∫_^)(t*e(−6*t)*d(t))=−t/6*e(−6*t)+1/6*(−1/6*e(−6*t))+C

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

(∫_^)(t*e(−6*t)*d(t))=−t/6*e(−6*t)−1/36*e(−6*t)+C

Final Answer

(∫_^)(t*e(−6*t)*d(t))=−1/6*t*e(−6*t)−1/36*e(−6*t)+C

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